## Linear Algebra and Its Applications, Exercise 2.3.22

Exercise 2.3.22. Given a vector space $V$ of dimension 7 and a subspace $W$ of $V$ of dimension 4, state whether the following are true or false:

1) You can create a basis for $V$ by adding three vectors to any set of vectors that is a basis for $W$.

2) You can create a basis for $W$ by removing three vectors from any set of vectors that is a basis for $V$.

Answer: 1) True. Suppose $w_1$, $w_2$, $w_3$, and $w_4$ are a basis for $W$. Per 2L (page 86) any linearly independent set in $V$ can be extended to a basis for $V$ by adding more vectors if necessary. The four vectors $w_1$ through $w_4$ are already linearly independent (since they are a basis) and hence can be extended by adding additional vectors to form a basis for $V$.

More specifically, we can find three vectors $v_1$, $v_2$, and $v_3$ such that a) the three vectors are not in $W$ (and hence are linearly independent of $w_1$ through $w_4$), and b) the three vectors are linearly independent of each other. The resulting seven vectors are linearly independent. Since the dimension of $V$ is 7 these seven linearly independent vectors must be a basis for $V$. (See exercise 2.3.15.)

2) False. Consider the vectors $v_1 = (1, 0, 0, 0, 0, 0, 0)$ through $v_7 = (0, 0, 0, 0, 0, 0, 1)$ with $v_i$ having a one in the $i^{th}$ position and zeros elsewhere. These vectors are linearly independent and span $V$ and hence are a basis for it.

Now suppose $W$ is the subspace of all vectors of the form $(a, a, b, b, c, c, d)$. The vectors $v_1$ through $v_6$ are not in the subspace $W$ and hence cannot be part of a basis for it. Thus it is not possible to remove three vectors from the basis set $v_1$ through $v_7$ and form a basis for $W$.

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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