## Linear Algebra and Its Applications, Exercise 2.3.12

Exercise 2.3.12.Suppose that the set of vectors $v_1$, $v_2$, $v_3$,  and $v_4$,  is a basis for $\mathbf{R}^4$ and that $W$ is a subspace of $\mathbf{R}^4$. Provide a counterexample to the conjecture that some subset of $v_1$, $v_2$, $v_3$,  and $v_4$ is necessarily a basis for $W$.

Answer: Suppose that $v_1$, $v_2$, $v_3$,  and $v_4$ are equal to the vectors $(1, 0, 0, 0)$, $(0, 1, 0, 0)$, $(0, 0, 1, 0)$, and $(0, 0, 0, 1)$ and suppose that $W$ is the subspace consisting of all vectors whose first two elements are equal to each other and whose last two elements are equal to each other; i.e., vectors in $W$ are of the form $(a, a, c, c)$. (It is fairly simple to verify that $W$ is in fact a subspace, so I omit that here.) In this case none of the vectors $v_1$, $v_2$, $v_3$,  and $v_4$ are in the subspace $W$ and thus cannot be part of a basis for $W$.

Instead we could use, for example, the vectors $(1, 1, 0, 0)$ and $(0, 0, 1, 1)$ as a basis for the subspace $W$, with any vector $w = (a, a, c, c)$ in the subspace expressible as $w = a \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + c \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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