Solved Problems for Chapter 5

Let Is a subspace? Explain.

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Let Is a subspace? Explain.

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Let and let Is a subspace? Explain.

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Let Is a subspace? Explain.

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If you have vectors in and the vectors are linearly independent, can it always be concluded they span Explain.

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If you have vectors in is it possible they are linearly independent? Explain.

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Suppose are subspaces of Let be all vectors which are in both and . Show that is a subspace also.

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Find the rank of the following matrix, and find a basis for the column spaces.

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Find a basis for .

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Find a basis for

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Suppose matrix has linearly independent columns. What is the null space of ?

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Let . Find a basis for . What is the dimension of ?

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Bibliography

Some of the problems come from the end of Chapter 4 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 227–232.

2024-09-06 23:26:34