Explain why B = {v1, v2, v3} is a Basis for ℝ^3, Find Coordinate Vector [x]B of x Relative to B

Let v1 = (1, 4, -5), v2 = (2, -3, -1), and v3 = (-4, 1, 7) (write as column vectors). Why does B = {v1, v2, v3} form a basis for ℝ^3? We need to show that B is a linearly independent set of vectors that spans ℝ^3. Both tasks are done by row reduction to reduced row echelon form (RREF). The fact that there is a pivot position in every column implies that B is a linearly independent set. The fact that there is a pivot position in every row implies that B is a spanning set for ℝ^3 (i.e., ℝ^3 = Span(B) = Span{v1, v2, v3}). If x = (3, 6, -2) (as a column vector), we can find the coordinate vector of x relative to the basis B, written [x]B, by solving a system of linear equations. "Linear Algebra and Its Applications", by David Lay: amzn.to/3sBMqKi.
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[@olivioarmandocordeirojunio8280]

Thanks a lot. Your explanations are very good.

[@billkinneymath]

Thanks!

[@vishalmittal3167]

Can you please upload lectures of using maxima software to solve first order and second order ordinary differential equations as you have uploaded for mathematica software..

[@billkinneymath]

Here's a video I did that might be helpful for you: th-cam.com/video/_hkJcZBCnAQ/w-d-xo.html