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Find a basis for each of the given subspaces and determine its dimension. a. V = Span _
Chapter 3, Problem 1(choose chapter or problem)
Find a basis for each of the given subspaces and determine its dimension.
a. \(V=\operatorname{Span}((1,2,3),(3,4,7),(5,-2,3)) \subset \mathbb{R}^{3}\)
b. \(V=\left\{\mathbf{x} \in \mathbb{R}^{4}: x_{1}+x_{2}+x_{3}+x_{4}=0, x_{2}+x_{4}=0\right\} \subset \mathbb{R}^{4}\)
c. \(V=(\operatorname{Span}((1,2,3)))^{\perp} \subset \mathbb{R}^{3}\)
d. \(V=\left\{\mathbf{x} \in \mathbb{R}^{5}: x_{1}=x_{2}, x_{3}=x_{4}\right\} \subset \mathbb{R}^{5}\)
Questions & Answers
(1 Reviews)
QUESTION:
Find a basis for each of the given subspaces and determine its dimension.
a. \(V=\operatorname{Span}((1,2,3),(3,4,7),(5,-2,3)) \subset \mathbb{R}^{3}\)
b. \(V=\left\{\mathbf{x} \in \mathbb{R}^{4}: x_{1}+x_{2}+x_{3}+x_{4}=0, x_{2}+x_{4}=0\right\} \subset \mathbb{R}^{4}\)
c. \(V=(\operatorname{Span}((1,2,3)))^{\perp} \subset \mathbb{R}^{3}\)
d. \(V=\left\{\mathbf{x} \in \mathbb{R}^{5}: x_{1}=x_{2}, x_{3}=x_{4}\right\} \subset \mathbb{R}^{5}\)
ANSWER:Step 1 of 6
A set \(V \subset {R^n}\) is called a subspace of \({R^n}\) if it satisfies all the following properties:
1. \(0 \in V\)
2. Whenever \(v \in V\) and \(c \in R\) then \(cv \in V\).
3. Whenever \(v,w \in V\), then \(v + w \in V\).
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