Are the Vectors S and T in the column of (ABC)

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SUMMARY

The discussion focuses on determining whether the vectors S=[1, 1, 0] and T=[-1, 0, 1] are in the column space of the product of three 3x3 matrices A, B, and C. Key insights include the use of Gaussian elimination to assess the rank of the matrix and the implications of adding vectors to a full rank set. The rank-nullity theorem is highlighted as a crucial tool for deducing the relationship between the rank of the column space and the null space.

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  • Understanding of linear algebra concepts, specifically vector spaces and linear combinations.
  • Proficiency in Gaussian elimination techniques for determining matrix rank.
  • Familiarity with the rank-nullity theorem and its implications in linear algebra.
  • Basic knowledge of matrix operations involving 3x3 matrices.
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  • Study Gaussian elimination methods for calculating the rank of matrices.
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  • Investigate the implications of adding vectors to a full rank set in matrix theory.
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify concepts related to vector spaces and matrix rank.

TomSavage
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I have no clue how to decode this question or do it but I was given the vectors S=[1 1 0] and T=[-1 0 1] and asked to determine whether or not they are in the column space of ABC when A, B and C are 3x3 matrices. My prof hinted to "think of rank and nullity", can someone please point me in the direction of the question cause as it stands right now I am lost, thanks.
 
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Hi Tom,

Hint 1: A vector is in the column space if it is a linear combination of the columns of the matrix.
Perhaps we can already visually see it.

Hint 2: We can find the rank of a matrix by Gaussian elimination and see how many vectors remain. Now add the vector in question, and repeat the elimination process. If we are left with an additional vector, it must have been independent of the original set, and is therefore not in the column space. Otherwise it was dependent, and is in the column space.

Hint 3: If we already have a full rank set of vectors, adding a vector cannot increase the rank. Then it must be dependent and in the column space.

Hint 4: If we know the rank of the null space, we can deduce the rank of the column space by the rank-nullity-theorem. That may help for hint 3 and hint 2.
 

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