What Are Vector Subspaces in Linear Algebra?

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Vector subspaces in linear algebra are formed by linear combinations of vectors, such as A, B, and C, expressed as xA + yB + zC, where x, y, and z are real numbers. The arrangement of these vectors is flexible; they can be represented in various formats, including a 2x2 matrix or a 4-element vector, depending on convenience. The generated space may indicate whether the vectors lie in the same plane or create a three-dimensional subspace within R^4. Understanding the span of these vectors is crucial for solving related problems. This foundational concept is essential for tackling assignments in linear algebra.
phy
Hi guys. I need some help with question #5 from my assignment. If someone can just tell me how to get the question started, it would be great. Thanks :smile:

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if a matrix is in the span of A,B,C, then it is written as a combination xA+yB+zC for some x,y,z in R.
 
Interesting problem. As was mentioned earlire, the space generated by A, B, and C will be xA+yB+zC. The format in which A, B, and C are written down is totally irrelevant - you can write them in a 2x2 square, but you could also write them down (in any order that's convenient) as a usual 4-element vector. Writing them in the more familiar form may make it easier. It's possible the three vectors lie in the same plane (you'd have to check), but it's more likely they generate a 3d subspace of R^4.
 

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