Linear Algebra and Its Applications, Exercise 2.3.23

Exercise 2.3.23. Let v_1 through v_9 be vectors in \mathbf{R}^7. Answer the following questions:

a) Are the nine vectors linearly independent? Not linearly independent? Might be linearly independent?

b) Do the nine vectors span \mathbf{R}^7? Not span \mathbf{R}^7? Might span \mathbf{R}^7?

c) Suppose the nine vectors are the columns of a matrix A. Does Ax = b have a solution? Not have a solution? Might have a solution?

Answer: a) We cannot have a set of 9 linearly independent vectors in a space like \mathbf{R}^7 that has dimension 7. So the vectors are not linearly independent.

b) The vectors might or might not span \mathbf{R}^7. For example, consider the set of vectors (1, 0, 0, 0, 0, 0, 0)(2, 0, 0, 0, 0, 0, 0), through (9, 0, 0, 0, 0, 0, 0). The nine vectors do not span \mathbf{R}^7 but rather span a subspace of dimension 1.

c) The matrix A would have nine columns but only seven rows, and would correspond to a system of seven linear equations with nine unknowns. This system could not have more than seven basic variables and thus would have at least two free variables. Since the free variables can take on any value the system Ax = 0 is guaranteed to have a solution (and in fact would have an infinite number of them) but the system Ax = b might or might not have a solution depending on the value of b.

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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One Response to Linear Algebra and Its Applications, Exercise 2.3.23

  1. Daniel says:

    This is a great way to review and refresh. Math, and math homework, were a long time ago but there are still some skills that would be handy to know. Thanks for a great resource.

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