MATH125 Lecture Notes - Lecture 24: Coordinate Vector

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MATH125 Full Course Notes
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MATH125 Full Course Notes
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The fundamental theorem of invertible matrices: version 2. Let a be a matrix of size n n. According to the fundamental theorem the vectors form a basis if the matrix with these vectors as its columns has rank 3, or equivalently 3 pivot positions. Thus we have three leading entries and we are done. The following theorem is also useful for applications. Let h be a p-dimensional subspace of rn. Any linearly independent set of exactly p elements in h is automatically a basis for h. also any set of p elements of h that span. Suppose that a matrix a of size 3 5 has three pivot columns. Since a has three pivot positions, it has three linearly inde- pendent columns, say a1, a2, a3, which span col a. Also, by the above basis theorem these columns a1, a2, a3 form a basis of r3 because r3 has dimension 3.

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