## 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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