TRUE OR FALSE For Exercises 5164, determine if the statementis true or false, and

For Exercises 16, find three vectors that are in the span of the given vectors. u1 =  2 6 ; u2 =  9 15

For Exercises 16, find three vectors that are in the span of the given vectors. u1 =  2 7 , u2 =  3 4

For Exercises 16, find three vectors that are in the span of the given vectors. u1 = 2 5 3  , u2 = 1 0 4 

For Exercises 16, find three vectors that are in the span of the given vectors. u1 = 0 5 2  , u2 = 1 2 6  , u3 = 6 7 2 

For Exercises 16, find three vectors that are in the span of the given vectors. u1 = 2 0 0  , u2 = 4 1 6  , u3 = 4 0 7 

For Exercises 16, find three vectors that are in the span of the given vectors. u1 = 0 1 3 0 , u2 = 1 8 5 2 , u3 = 12 1 1 0

For Exercises 712, determine if b is in the span of the other given vectors. If so, write b as a linear combination of the other vectors. a1 =  3 5 , b =  9 15

For Exercises 712, determine if b is in the span of the other given vectors. If so, write b as a linear combination of the other vectors. a1 =  10 15 , b =  30 45

For Exercises 712, determine if b is in the span of the other given vectors. If so, write b as a linear combination of the other vectors. a1 = 4 2 10 , b = 2 1 5 

For Exercises 712, determine if b is in the span of the other given vectors. If so, write b as a linear combination of the other vectors. a1 = 1 3 1  , a2 = 2 3 6  , b = 6 9 2 

For Exercises 712, determine if b is in the span of the other given vectors. If so, write b as a linear combination of the other vectors. a1 = 1 4 3  , a2 = 2 8 7  , b = 10 8 7 

For Exercises 712, determine if b is in the span of the other given vectors. If so, write b as a linear combination of the other vectors. a1 = 3 1 2 1 , a2 = 4 2 3 3 , b = 0 10 1 5

In Exercises 1316, find A, x, and b such that Ax = b corresponds to the given linear system. 2x1 + 8x2 4x3 = 10 x1 3x2 + 5x3 = 4

In Exercises 1316, find A, x, and b such that Ax = b corresponds to the given linear system. 2x1 + 5x2 10x3 = 4 x1 2x2 + 3x3 = 1 7x1 17x2 + 34x3 = 16

In Exercises 1316, find A, x, and b such that Ax = b corresponds to the given linear system. x1 x2 3x3 x4 = 1 2x1 + 2x2 + 6x3 + 2x4 = 1 3x1 3x2 + 10x3 = 5

In Exercises 1316, find A, x, and b such that Ax = b corresponds to the given linear system. . 5x1 + 9x2 = 13 3x1 5x2 = 9 x1 2x2 = 2

In Exercises 1720, find an equation involving vectors that corresponds to the given linear system. 5x1 + 7x2 2x3 = 9 x1 5x2 4x3 = 2

In Exercises 1720, find an equation involving vectors that corresponds to the given linear system. 4x1 5x2 3x3 = 0 3x1 + 4x2 + 2x3 = 1 6x1 13x2 + 7x3 = 2

In Exercises 1720, find an equation involving vectors that corresponds to the given linear system. 4x1 2x2 3x3 + 5x4 = 12 5x2 + 7x3 + 3x4 = 6 3x1 + 8x2 + 2x3 x4 = 2

In Exercises 1720, find an equation involving vectors that corresponds to the given linear system. 4x1 9x2 = 11 2x1 + 4x2 = 9 x1 7x2 = 2

In Exercises 2124, determine if the columns of the given matrix span R2. 15 6 5 2

In Exercises 2124, determine if the columns of the given matrix span R2. 4 12 2 6

In Exercises 2124, determine if the columns of the given matrix span R2. 210 6 3 1 2

In Exercises 2124, determine if the columns of the given matrix span R2. 105 227

In Exercises 2528, determine if the columns of the given matrix span R3. 310 5 2 1 4 4 3 

In Exercises 2528, determine if the columns of the given matrix span R3.  1 28 2 37 3 1 1

In Exercises 2528, determine if the columns of the given matrix span R3. 2 1 3 5 14 26 03 33

In Exercises 2528, determine if the columns of the given matrix span R3. 4 71 2 0 03 8 5 1 1 4 

In Exercises 2934, a matrix A is given. Determine if the system Ax = b (where x and b have the appropriate number of components) has a solution for all choices of b. A = 3 4 4 2 3

In Exercises 2934, a matrix A is given. Determine if the system Ax = b (where x and b have the appropriate number of components) has a solution for all choices of b. A = 9 21 6 14 3

In Exercises 2934, a matrix A is given. Determine if the system Ax = b (where x and b have the appropriate number of components) has a solution for all choices of b. A =  8 1 0 1 3 2

In Exercises 2934, a matrix A is given. Determine if the system Ax = b (where x and b have the appropriate number of components) has a solution for all choices of b. A =  1 1 2 2 3 1 105

In Exercises 2934, a matrix A is given. Determine if the system Ax = b (where x and b have the appropriate number of components) has a solution for all choices of b. A = 321 1 1 1 5 4 3

In Exercises 2934, a matrix A is given. Determine if the system Ax = b (where x and b have the appropriate number of components) has a solution for all choices of b. A = 2 3 0 012 5 3 9 309

For Exercises 3538, find a vector of matching dimension that is not in the given span. span 1 2 , 3 6 36.

For Exercises 3538, find a vector of matching dimension that is not in the given span. span3 1 , 6 2 37.

For Exercises 3538, find a vector of matching dimension that is not in the given span. span 1 3 2  ,  2 1 1 

For Exercises 3538, find a vector of matching dimension that is not in the given span. span1 2 1  ,  3 1 1  , 1 5 1 

Find all values of h such that the vectors {a1, a2} span R2, where a1 = 2 4 , a2 = h 6

Find all values of h such that the vectors {a1, a2} span R2, where a1 = 3 h , a2 = 5 4

Find all values of h such that the vectors {a1, a2, a3} span R3, where a1 = 2 4 5  , a2 =  h 8 10 , a3 = 1 2 6 

Find all values of h such that the vectors {a1, a2, a3} span R3, where a1 = 1 h 7  , a2 =  4 2 5  , a3 =  1 3 2 

FIND AN EXAMPLE For Exercises 4350, find an example that meets the given specifications. Four distinct nonzero vectors that span R3.

FIND AN EXAMPLE For Exercises 4350, find an example that meets the given specifications. Four distinct nonzero vectors that span R4.

FIND AN EXAMPLE For Exercises 4350, find an example that meets the given specifications. Four distinct nonzero vectors that do not span R3.

FIND AN EXAMPLE For Exercises 4350, find an example that meets the given specifications. Four distinct nonzero vectors that do not span R4.

FIND AN EXAMPLE For Exercises 4350, find an example that meets the given specifications. Two vectors u1 and u2 in R3 that span the set of all vectors of the form v = (v1, v2, 0).

FIND AN EXAMPLE For Exercises 4350, find an example that meets the given specifications. Three vectors u1, u2, and u3 inR4 that span the set of all vectors of the form v = (0, v2, v3, v4).

FIND AN EXAMPLE For Exercises 4350, find an example that meets the given specifications. Two vectors u1 and u2 in R3 that span the set of all vectors of the form v = (v1, v2, v3) where v1 + v2 + v3 = 0.

FIND AN EXAMPLE For Exercises 4350, find an example that meets the given specifications. Three vectors u1, u2, and u3 inR4 that span the set of all vectors of the form v = (v1, v2, v3, v4) where v1 + v2 + v3 + v4 = 0.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If m < n, then a set of m vectors cannot span Rn.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If a set of vectors includes 0, then it cannot span Rn.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. Suppose A is a matrix with n rows and m columns. If n < m, then the columns of A span Rn.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. Suppose A is a matrix with n rows and m columns. If m < n, then the columns of A span Rn.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If A is a matrix with columns that span Rn, then Ax = 0 has nontrivial solutions.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If A is a matrix with columns that span Rn, then Ax = b has a solution for all b in Rn.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If {u1, u2, u3} spans R3, then so does {u1, u2, u3, u4}.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If {u1, u2, u3} does not span R3, then neither does {u1, u2, u3, u4}.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If {u1, u2, u3, u4} spans R3, then so does {u1, u2, u3}.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If {u1, u2, u3, u4} does not span R3, then neither does {u1, u2, u3}.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If u4 is a linear combination of {u1, u2, u3}, then span{u1, u2, u3, u4} = span{u1, u2, u3}.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If u4 is a linear combination of {u1, u2, u3}, then span{u1, u2, u3, u4} = span{u1, u2, u3}.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If u4 is not a linear combination of {u1, u2, u3}, then span{u1, u2, u3, u4} = span{u1, u2, u3}.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. If u4 is not a linear combination of {u1, u2, u3}, then span{u1, u2, u3, u4} = span{u1, u2, u3}.

TRUE OR FALSE For Exercises 5164, determine if the statement is true or false, and justify your answer. Which of the following sets of vectors in R3 can possibly span R3? Justify your answer. (a) {u1} (b) {u1, u2} (c) {u1, u2, u3} (d) {u1, u2, u3, u4}

Which of the following sets of vectors in R3 cannot possibly span R3? Justify your answer. (a) {u1} (b) {u1, u2} (c) {u1, u2, u3} (d) {u1, u2, u3, u4}

Prove that if c is a nonzero scalar, then span{u} = span{cu}.

Prove that ifc1 and c2 are nonzero scalars, then span{u1, u2} = span{c1u1, c2u2}.

Suppose that S1 are S2 are two finite sets of vectors, and that S1 is a subset of S2. Prove that the span of S1 is a subset of the span of S2.

Prove that if span{u1, u2} = R2, then span{u1 + u2, u1 u2} = R2.

Prove that if span{u1, u2, u3} = R3, then span{u1 + u2, u1 + u3, u2 + u3} = R3

Suppose that{u1, ... , um}is a subset ofRn, withm > n. Prove that if b is in span{u1, ... , um}, then there are infinitely many ways to express b as a linear combination of {u1, ... , um}.

Prove Theorem 2.8.

Prove Theorem 2.10.

C For Exercises 7578, determine if the claimed equality is true or false. span 3 2 1  , 1 5 4  , 2 3 0  = R3

C For Exercises 7578, determine if the claimed equality is true or false. span 1 1 2  , 3 4 1  , 4 6 6  = R3

C For Exercises 7578, determine if the claimed equality is true or false. span 4 0 2 3 , 7 4 6 7 , 1 3 2 1 , 3 2 0 2 = R4

C For Exercises 7578, determine if the claimed equality is true or false. span 3 2 1 0 , 8 5 9 7 , 1 6 2 3 , 2 1 3 5 = R4