?Sizing Them Up For each of 57 and 58, determine the dimension and find a basis for the

The Spin on Spans Determine whether the vectors in the set S span the vector space V. \(\mathbb{V}=\mathbb{R}^{2} ; \quad S=\{[0,0],[1,1]\}\)

The Spin on Spans Determine whether the vectors in the set S span the vector space V. \(\mathbf{V}=\mathbb{R}^{3} ; \quad S=\{[1,0,0],[0,1,0],[2,3,1]\}\)

The Spin on Spans Determine whether the vectors in the set S span the vector space V. \(\mathbf{V}=\mathbb{R}^{3} ; \quad S=\{[1,0,-1],[2,0,4],[-5,0,2],[0,0,1]\}\)

The Spin on Spans Determine whether the vectors in the set S span the vector space V. \(\mathbf{V}=\mathbb{P}_{2} ; \quad S=\left\{1, t+1, t^{2}-2 t+3\right\}\)

The Spin on Spans Determine whether the vectors in the set S span the vector space V. \(\mathbf{V}=\mathbb{P}_{2} ; \quad S=\left\{t+1, t^{2}+1, t^{2}-t\right\}\)

The Spin on Spans Determine whether the vectors in the set S span the vector space V. \(\begin{array}{l} \mathbf{V}=\mathbf{M}_{22} \\ S=\left\{\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 1 \end{array}\right],\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 1 \end{array}\right]\right\} \end{array} \)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{R}^{2} ; \quad S=\{[1,-1],[-1,1]\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{R}^{2} ; \quad S=\{[1,1],[1,-1]\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{R}^{3} ; \quad S=\{[1,0,0],[1,1,0],[1,1,1]\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{R}^{3} ; \quad S=\{[2,-1,4],[4,-2,8]\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V=\mathbb{R}^{3} ; \quad S=\{[1,1,8],[-3,4,2],[7,-1,3]\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{P}_{1} ; \quad S=\{1, t\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{P}_{1} ; \quad S=\{1+t, 1-t\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{P}_{2} ; \quad S=\{t, 1-t\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{P}_{2} ; \quad S=\left\{1+t, 1-t, t^{2}\right\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{P}_{2} ; \quad S=\left\{t+3, t^{2}-1,2 t^{2}-t-5\right\}\)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\begin{array}{l} \mathbf{V}=\mathbb{D}_{22} \text { (the diagonal } 2 \times 2 \text { matrices); } \\ S=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\} \end{array} \)

Independence Day Decide whether the set S is a linearly independent subset of the given vector space V. \(\mathbf{V}=\mathbb{D}_{22} ; S=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right],\left[\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right]\right\} \)

Function Space Dependence Determine whether each subset S of C(?) in Problems 19-25 is or is not linearly independent. \(S=\left\{e^{t}, e^{-t}\right\}\)

Function Space Dependence Determine whether each subset S of C(?) in Problems 19-25 is or is not linearly independent. \(S=\left\{e^{t}, t e^{t}, t^{2} e^{t}\right\}\)

Function Space Dependence Determine whether each subset S of C(?) in Problems 19-25 is or is not linearly independent. \(S=\{\sin t, \sin 2 t, \sin 3 t\}\)

Function Space Dependence Determine whether each subset S of C(?) in Problems 19-25 is or is not linearly independent. \(S=\left\{1, \sin ^{2} t, \cos ^{2} t\right\}\)

Function Space Dependence Determine whether each subset S of C(?) in Problems 19-25 is or is not linearly independent. \(S=\left\{1, t-1,(t-1)^{2}\right\}\)

Function Space Dependence Determine whether each subset S of C(?) in Problems 19-25 is or is not linearly independent. \(S=\left(e^{t}, e^{-t}, \cosh t\right)\)

Function Space Dependence Determine whether each subset S of C(?) in Problems 19-25 is or is not linearly independent. \(S=\left\{\sin ^{2} t, 4, \cos 2 t\right\}\)

Independence Testing Determine whether or not the set of functions given in each of Problems 25-29 is linearly independent on \((-\infty, \infty,)\). \(\left\{\left[\begin{array}{l} e^{t} \\ e^{t} \end{array}\right],\left[\begin{array}{c} 2 e^{2 t} \\ e^{t} \end{array}\right]\right\} \)

Independence Testing Determine whether or not the set of functions given in each of Problems 25-29 is linearly independent on \((-\infty, \infty,)\). \(\left\{\left[\begin{array}{l} \sin t \\ \cos t \end{array}\right],\left[\begin{array}{r} \cos t \\ -\sin t \end{array}\right]\right\} \)

Independence Testing Determine whether or not the set of functions given in each of Problems 25-29 is linearly independent on \((-\infty, \infty,)\). \(\left\{\left[\begin{array}{r} e^{t} \\ 2 e^{t} \\ e^{t} \end{array}\right],\left[\begin{array}{c} e^{-t} \\ 2 e^{-t} \\ e^{t} \end{array}\right],\left[\begin{array}{c} e^{2 t} \\ 3 e^{2 t} \\ e^{2 t} \end{array}\right]\right\} \)

Independence Testing Determine whether or not the set of functions given in each of Problems 25-29 is linearly independent on \((-\infty, \infty,)\). \(\left\{\left[\begin{array}{r} e^{-t} \\ -4 e^{-t} \\ e^{-t} \end{array}\right],\left[\begin{array}{c} e^{-t} \\ 0 \\ -e^{-t} \end{array}\right],\left[\begin{array}{c} 2 e^{8 t} \\ e^{8 t} \\ 2 e^{8 t} \end{array}\right]\right\} \)

Twins? Is there any difference between the vector space spanned by the set \({cos t,sin t)\) and the vector space spanned by the set \(S=\{\cos t+\sin t, \cos t-\sin t\}\)?

A Questionable Basis Is the set \(\left\{\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ -1 \end{array}\right]\right\} \) a basis for \(\mathbb{R}^{3}\)? If not, change one of the vectors to form a basis.

Wronskian Suppose that f and g are differentiable functions on the unit interval l. The Wronskian of f and g is given by \(\begin{aligned} W[f, g](t) &=\left|\begin{array}{ll} f(t) & g(t) \\ f^{\prime}(t) & g^{\prime}(t) \end{array}\right| \\ &=f(t) g^{\prime}(t)-f^{\prime}(t) g(t) \end{aligned} \) Show that if \(W[f, g](t) \neq 0\) for all t in I. then f and g are linearly independent. HINT: If \(c_{1} f(t)+c_{2} g(t)=0\), then \(c_{1} f^{\prime}(t)+c_{2} g^{\prime}(t)=0\); use Cramer’s Rule to solve for \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\).

Zero Wronskian Does Not Imply Linear Dependence We saw that a nonzero Wronskian implies linear independence but that the converse is not true. (See fa.ample 12.) Another example follows: (a) Show that the Wronskian is zero for lhe functions \(f(t)=t^{2} \quad \text { and } \quad g(t)=t|t|=\left\{\begin{array}{ll} t^{2} & \text { if } t \geq 0 \\ -t^{2} & \text { if } t<0 \end{array}\right. \) (b) Show lhal the functions f and g are not linearly dependent but independent on (\(-\infty, \infty\)).

Linearly Independent Exponentials Use the Wronskian to show that (eat, ebt) I is a linearly independent set if and only if \(a \neq b\).

Looking Ahead Use the Wronskian to show that {et, tet} is linearly independent on \(\mathbb{R}\).

Revisiting Linear Independence Use the Wronskian to show that the set {\(e^{t}, 5^{e-t}, e^{3 t}\)} is linearly independent. (See Ex.ample 7.)

Independence Checking Use the Wronskian to check the subsets of C(\(\mathbb{R}\)) in Problems 37-42 for linear independence. \(\{5, \cos t, \sin t\}\)

Independence Checking Use the Wronskian to check the subsets of C(\(\mathbb{R}\)) in Problems 37-42 for linear independence. \(\left\{e^{t}, e^{-t}, 1\right\}\)

Independence Checking Use the Wronskian to check the subsets of C(\(\mathbb{R}\)) in Problems 37-42 for linear independence. \(\left\{2+t, 2-t, t^{2}\right\}\)

Independence Checking Use the Wronskian to check the subsets of C(\(\mathbb{R}\)) in Problems 37-42 for linear independence. \(\left\{3 t^{2}-4,2 t, t^{2}-1\right\}\)

Independence Checking Use the Wronskian to check the subsets of C(\(\mathbb{R}\)) in Problems 37-42 for linear independence. \(\{\cosh t, \sinh t\}\)

Independence Checking Use the Wronskian to check the subsets of C(\(\mathbb{R}\)) in Problems 37-42 for linear independence. \(\left\{e^{t} \cos t, e^{t} \sin t\right\}\)

Getting on Base in \(\mathbb{R}^2\) Determine whether the set given in each of Problems 43-48 is a basis for \(\mathbb{R}^2\) Justify your answers. \(\{[1,1]\}\)

Getting on Base in \(\mathbb{R}^2\) Determine whether the set given in each of Problems 43-48 is a basis for \(\mathbb{R}^2\) Justify your answers. \(\{[1,2],[2,1]\}\)

Getting on Base in \(\mathbb{R}^2\) Determine whether the set given in each of Problems 43-48 is a basis for \(\mathbb{R}^2\) Justify your answers. \(\{[-1,-1],[1,1]\}\)

Getting on Base in \(\mathbb{R}^2\) Determine whether the set given in each of Problems 43-48 is a basis for \(\mathbb{R}^2\) Justify your answers. \(\{[1,0],[1,1]\}\)

Getting on Base in \(\mathbb{R}^2\) Determine whether the set given in each of Problems 43-48 is a basis for \(\mathbb{R}^2\) Justify your answers. \(\{[1,0],[0,1],[1,1]\}\)

Getting on Base in \(\mathbb{R}^2\) Determine whether the set given in each of Problems 43-48 is a basis for \(\mathbb{R}^2\) Justify your answers. \(\{[0,0],[1,1],[2,2],[-1,-1]\}\)

The Base for the Space Determine whether or not the set S in each of Problems 49-56 is a basis for the specified vector space V. \(\mathbf{V}=\mathbb{R}^{3} ; \quad S=\{[1,0,0],[0,1,0]\}\)

The Base for the Space Determine whether or not the set S in each of Problems 49-56 is a basis for the specified vector space V. \(\mathbf{V}=\mathbb{R}^{3} ; \quad S=\{[1,0,1],[1,1,0],[0,1,1]\}\)

The Base for the Space Determine whether or not the set S in each of Problems 49-56 is a basis for the specified vector space V. \(\mathbf{V}=\mathbb{R}^{3} ; \quad S=\{[1,0,0],[0,1,0],[0,1,1],[1,1,0]\}\)

The Base for the Space Determine whether or not the set S in each of Problems 49-56 is a basis for the specified vector space V. \(\mathbf{V}=\mathbb{P}_{2} ; \quad S=\left\{t^{2}+3 t+1, t^{2}-2 t+4\right\}\)

The Base for the Space Determine whether or not the set S in each of Problems 49-56 is a basis for the specified vector space V. \(\mathbf{V}=\mathbb{P}_{3} ; \quad S=\left\{t^{2}, t+3, t^{3}+4, t-1, t^{2}-5 t+1\right\}\)

The Base for the Space Determine whether or not the set S in each of Problems 49-56 is a basis for the specified vector space V. \(\mathbf{V}=\mathbb{P}_{4} ; \quad S=\left\{t^{4}, t+3, t^{3}+4, t-1, t^{2}-5 t+1\right\}\)

The Base for the Space Determine whether or not the set S in each of Problems 49-56 is a basis for the specified vector space V. \(\begin{array}{l} \mathbf{V}=\mathbf{M}_{22} ; \\ S=\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right] \cdot\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right] \cdot\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right] \cdot\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\right\} \end{array} \)

The Base for the Space Determine whether or not the set S in each of Problems 49-56 is a basis for the specified vector space V. \(\begin{aligned} \mathbf{V}=& \mathbf{M}_{23} \\ S=&\left\{\left[\begin{array}{lll} 1 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right],\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right],\left[\begin{array}{lll} 0 & 0 & 0 \\ 1 & 0 & 1 \end{array}\right]\right.\\ &\left.\left[\begin{array}{lll} 0 & 0 & 0 \\ 1 & 1 & 0 \end{array}\right],\left[\begin{array}{lll} 0 & 0 & 0 \\ 1 & 1 & 1 \end{array}\right]\right\} \end{aligned} \)

Sizing Them Up For each of Problems 57 and 58, determine the dimension and find a basis for the subspace W of the vector space V. \(\mathbf{V}=\mathbb{R}^{3} ; \quad \mathbf{w}=\left\{\left[x_{1}, x_{2}, x_{3}\right] \mid x_{1}+x_{2}+x_{3}=0\right\}\)

Sizing Them Up For each of Problems 57 and 58, determine the dimension and find a basis for the subspace W of the vector space V. \(\left.\mathbf{V}=\mathbb{R}^{4} ; \quad \mathbf{W}=\| x_{1}, x_{2}, x_{3}, x_{4}\right]\left|x_{1}+x_{3}=0, x_{2}=x_{4}\right\rangle\)

Polynomial Dimensions Find the dimension of the subspace of \(\mathbb{P}_{3}\). spanned by the subset in Problems 59-61. \(\{t, t-1\}\)

Polynomial Dimensions Find the dimension of the subspace of \(\mathbb{P}_{3}\). spanned by the subset in Problems 59-61. \(\left\{t, t-1, t^{2}+1\right\}\)

Polynomial Dimensions Find the dimension of the subspace of \(\mathbb{P}_{3}\). spanned by the subset in Problems 59-61. \(\left\{t^{2}, t^{2}-t-1, t+1\right\}\)

Solution Basis Determine a basis for the solution set (a subspace of ?3) of the system \(\begin{array}{r} x+y-z=0 \\ y-5 z=0 \end{array} \)

Solution Spaces for Linear Algebraic Systems In Problems 63 and 64, determine bases and dimension for the solution spaces for the homogeneous systems (as given in Section 3.5, Problems 61 and 62). \(\begin{array}{r} 2 x_{1}-2 x_{2}+4 x_{3}-2 x_{4}=0 \\ 2 x_{1}+x_{2}+7 x_{3}+4 x_{4}=0 \\ x_{1}-4 x_{2}-x_{3}+7 x_{4}=0 \\ 4 x_{1}-12 x_{2}-20 x_{4}=0 \end{array} \)

Solution Spaces for Linear Algebraic Systems In Problems 63 and 64, determine bases and dimension for the solution spaces for the homogeneous systems (as given in Section 3.5, Problems 61 and 62). \(\begin{array}{r} 3 x_{1}+6 x_{3}+3 x_{4}+9 x_{5}=0 \\ x_{1}+3 x_{2}-4 x_{3}-8 x_{4}+3 x_{5}=0 \\ x_{1}-6 x_{2}+14 x_{3}+19 x_{4}+3 x_{5}=0 \end{array} \)

DE Solution Spaces For each of the differential equations given in Problems 65-70, consider their solution sets. (a) Does the solution set fonn a subspace of the specified larger space? If not, aplain. (b) If the solution set is a subspace, find a basis and determine the dimension. \(d^{n} y / d t^{n}=0, \quad \mathcal{C}^{n}(\mathbb{R})\)

DE Solution Spaces For each of the differential equations given in Problems 65-70, consider their solution sets. (a) Does the solution set fonn a subspace of the specified larger space? If not, aplain. (b) If the solution set is a subspace, find a basis and determine the dimension. \(y^{\prime}-2 y=0, \quad \mathcal{C}^{\prime}(\mathbb{R})\)

DE Solution Spaces For each of the differential equations given in Problems 65-70, consider their solution sets. (a) Does the solution set fonn a subspace of the specified larger space? If not, aplain. (b) If the solution set is a subspace, find a basis and determine the dimension. \(y^{\prime}-2 t y=0, \quad \mathcal{C}^{\prime}\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

DE Solution Spaces For each of the differential equations given in Problems 65-70, consider their solution sets. (a) Does the solution set fonn a subspace of the specified larger space? If not, aplain. (b) If the solution set is a subspace, find a basis and determine the dimension. \(y^{\prime}+(\tan t) y=0, \quad \mathcal{C}^{\prime}(\mathbb{R})\)

DE Solution Spaces For each of the differential equations given in Problems 65-70, consider their solution sets. (a) Does the solution set fonn a subspace of the specified larger space? If not, aplain. (b) If the solution set is a subspace, find a basis and determine the dimension. \(y^{\prime}+y^{2}=0, \quad \mathcal{C}^{\prime}(\mathbb{R})\)

DE Solution Spaces For each of the differential equations given in Problems 65-70, consider their solution sets. (a) Does the solution set fonn a subspace of the specified larger space? If not, aplain. (b) If the solution set is a subspace, find a basis and determine the dimension. \(y^{\prime}+(\cos t) y=0, \quad \mathcal{C}^{\prime}(\mathbb{R})\)

Bases for Subspaces of \(\mathbb{R}^n\) In Problems 71-73, find a basis and the dimension of the given subspaces of \(\mathbb{R}^n\). \(\{[a, 0, b, a-b+c] \mid a, b, c \in \mathbb{R}\} \text { as a subspace of } \mathbb{R}^{4}\)

Bases for Subspaces of \(\mathbb{R}^n\) In Problems 71-73, find a basis and the dimension of the given subspaces of \(\mathbb{R}^n\). \(\{[a, a-b, 2 a+3 b] \mid a, b, \in \mathbb{R}\} \text { as a subspace of } \mathbb{R}^{3}\)

Bases for Subspaces of \(\mathbb{R}^n\) In Problems 71-73, find a basis and the dimension of the given subspaces of \(\mathbb{R}^n\). \(\{[x+y+z, x+y, 4 z, 0] \mid x, y, z \in \mathbb{R}\} \text { as a subspace of } \mathbb{R}^{4}\)

Two-by-Two Basis Show that \(\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 1 \end{array}\right]\right\} \) is a linearly independent set in \(\mathrm{M}_{22}\). Add another vector to the set to make it a basis for \(\mathrm{M}_{22}\).

Basis for Zero Trace Matrices Show that \(\left\{\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right]\right\} \) is a basis for the subspace of all 2 x 2 matrices in \(\mathrm{M}_{22}\) with zero trace.

Hyperplane Basis Find a basis for the following hyperplane in \(\mathbb{R}^4: x + 3y -2z + 6w = 0\).

Symmetric Matrices Find the dimension and exhibit a basis for the subspace of all symmetric matrices in \(\mathrm{M}_{22}\).

Making New Bases from Old In Problems 78-80, a vector space and a basis will be given. Construct a different basis from the given basis. Make them substantially different, so that at least two vectors are not multiples of vectors in the original basis. \(\{\bar{i}, \bar{j}, \bar{k}\} \text { in } \mathbb{R}^{3}\)

Making New Bases from Old In Problems 78-80, a vector space and a basis will be given. Construct a different basis from the given basis. Make them substantially different, so that at least two vectors are not multiples of vectors in the original basis. \(\left\{\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right]\right\} \text { in } \mathbb{D} \), the space of all diagonal matrices

Making New Bases from Old In Problems 78-80, a vector space and a basis will be given. Construct a different basis from the given basis. Make them substantially different, so that at least two vectors are not multiples of vectors in the original basis. \(\{\sin t, \cos t\} \text { in }\left\{y \in \mathcal{C}^{\prime \prime}(\mathbb{R})\left|y^{\prime \prime}-y=0\right|\right.\)

Basis for \(\mathbb{P}_{2}\) Do the vectors \(\left\{t^{2}+t+1, t+1,1\right\}\) form a basis for \(\mathbb{P}_{2}\)? If so, represent the polynomial \(3 t^{2}+2 t+1\) in tenns of this basis.

True/False Questions If false, give a counterexample or a brief explanation. (a) A solution set of a homogeneous system of linear algebraic equations, given by \(\mathbf{w}=\left\{s\left[\begin{array}{r} 3 \\ 0 \\ 0 \\ -1 \end{array}\right]+t\left[\begin{array}{l} 0 \\ 2 \\ 1 \\ 1 \end{array}\right] \mid s, t \in \mathbb{R}\right\} \) is a subspace of \(\mathbb{R}^{4}\). True or false? (b) The dimension of W is 4. True or false? (c) A basis for, W is \(\{[3,0],[0,2],[0,1],[-1,1]\}\). True or false?

Essay Question3 Consider the homogeneous system of linear equations \(\begin{aligned} x_{1}+x_{2}+x_{3}-x_{4} &=0 \\ 3 x_{1}+3 x_{2}+x_{3}-5 x_{4} &=0 \\ 4 x_{1}+4 x_{2}+2 x_{3}-6 x_{4} &=0 \end{aligned} \) and its solution set \(\mathbf{W}=\left\{s\left[\begin{array}{r} -1 \\ 1 \\ 0 \\ 0 \end{array}\right]+t\left[\begin{array}{r} 2 \\ 0 \\ -1 \\ 1 \end{array}\right] \mid s, t \in \mathbb{R}\right\} \) Write a paragraph or two describing the solution set W of the system of equations. Write a clear exposition that includes appropriate use of the words linear combination, span, subspace, vector space, linearly independent, basis, and dimension in describing the solution set.

Convergent Sequence Space4 Discuss the set V of all convergent sequences as a vector space, where the addition of "vectors" and multiplication by a scalar are defined by \(\left\{a_{n}\right\}+\left\{b_{n}\right\}=\left\{a_{n}+b_{n}\right\} \quad \text { and } \quad c\left\{a_{n}\right\}=\left\{c a_{n}\right\}\) Describe the zero element and additive inverse elements for this vector space. Make a conjecture about the dimension of V and try to back it up. Give at least one example of a non-trivial subspace and discuss its dimension.

Cosets in \(\mathbb{R}^{3}\) Using the following definition, in Problems 85 and 86, find the W-cosets for the given vectors \(\bar{v}\) and give a graphical description of each. Cosets If W is a subspace of vector space \(\mathbb{R}^{n}\) that includes the origin, and \(\bar{v}\) is a vector \(\mathbb{R}^{n}\), then the W-coset of \(\bar{v}\), denoted \(\bar{v}\) + W, is the set defined by \(\bar{v} + W =\) {\(\bar{v} + \bar{w} | \ \bar{w} \ is \ in \ W \)} Thus, in \(\mathbb{R}^{3}\), if W is a plane passing through the origin, then the coset \(\bar{v} + W\) is a plane parallel to W passing through \(\bar{v}\) \(\mathbf{w}=\left\{\left[x_{1}, x_{2}, x_{3}\right] \mid x_{1}+x_{2}+x_{3}=0\right\} ; \bar{\mathbf{v}}=(0,0,1)\)

Cosets in \(\mathbb{R}^{3}\) Using the following definition, in Problems 85 and 86, find the W-cosets for the given vectors \(\bar{v}\) and give a graphical description of each. Cosets If W is a subspace of vector space \(\mathbb{R}^{n}\) that includes the origin, and \(\bar{v}\) is a vector \(\mathbb{R}^{n}\), then the W-coset of \(\bar{v}\), denoted \(\bar{v}\) + W, is the set defined by \(\bar{v} + W =\) {\(\bar{v} + \bar{w} | \ \bar{w} \ is \ in \ W \)} Thus, in \(\mathbb{R}^{3}\), if W is a plane passing through the origin, then the coset \(\bar{v} + W\) is a plane parallel to W passing through \(\bar{v}\) \(\mathbf{W}=\left\{\left[x_{1}, x_{2}, x_{3}\right] \mid x_{3}=0\right\} ; \quad \bar {\mathbf{v}}=(1,1,1)\)

More Cosets A sin Problems 85 and 86, describe the nature of a coset if the subspace W is a line through the origin. Illustrate for \(\begin{aligned} \mathbf{W} &=\left\{\left[x_{1}, x_{2}, x_{3}\right] \mid x_{1}=t, x_{2}=3 t, x_{3}=2 t\right\} \quad \text { and } \\ \bar{\mathbf{v}} &=(1,-2,1) . \end{aligned} \)

Line in Function Space Interpret the general solution of the differential equation \(y^{\prime}+2 y=e^{-2 t}\) as a line in a suitable function space.

Mutual Orthogonality Prove that nonzero mutually orthogonal vectors in a vector space V are automatically linearly independent.

Suggested Journal Entry I Discuss the distinction between finite and finite-dimensional for vector spaces. Can a vector space contain only a finite number of vectors? Explain.

Suggested Journal Entry II Does the set of even functions (where \(f (t) = f (-t)\) on the interval \((-\infty, \infty)\)) constitute a vector space? What about the set of polynomials of even degree? What is the relationship between these two families? How big are they? ________________ Equation Transcription: (-, ) Text Transcription: f (t) = f (-t) (-infinity, infinity)

Suggested Journal Entry III Look at Examples \(9\) and \(10\) and the paragraph about span and basis. Then consider the case of two vector functions with three components each. What size system would you need to determine whether the vectors were linearly independent? Consider the same question for the case of three vector functions of two components each ________________ Equation Transcription: Text Transcription: 9 10