Linear Algebra and Its Applications, Exercise 2.3.11

Exercise 2.3.11. Consider the subspace of $\mathbf{R}^3$ consisting of all vectors whose first two components are equal. Find two different bases for this subspace.

Answer: All vectors in the subspace are of the form $(a, a, c)$ . One basis for the subspace consists of the vectors $(1, 1, 0)$ and $(0, 0, 1)$ with any vector $v = (a, a, c)$ in the subspace expressible as

$v = a \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} + c \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$

Another basis for the subspace consists of the vectors $(1, 1, 1)$ and $(0, 0, -1)$ with any vector $v = (a, a, c)$ in the subspace expressible as

$v = a \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} + (a - c) \begin{bmatrix} 0 \\ 0 \\ -1 \end{bmatrix}$

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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