## 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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