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Matrices and Linear Spaces

Quick questions on Linear spaces explained: H2 Further Mathematics

4short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is rank of a matrix?
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The rank of a matrix is the number of linearly independent columns (equivalently independent rows), found by reducing to row-echelon form and counting the non-zero rows. It is the dimension of the column space, the subspace spanned by the columns.
What is q1?
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State the condition for vectors v1,,vk\mathbf{v}_1, \dots, \mathbf{v}_k to be linearly independent. [2 marks]
What is q2?
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A 44-column matrix has rank 33. State its nullity. [1 mark]
What is q3?
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Why is a line not passing through the origin not a subspace? [2 marks]

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