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

In summary, the conversation discusses how to determine whether or not given vectors are in the column space of a matrix. The hints provided by the professor suggest using the concepts of rank and nullity to solve the problem. Hint 1 suggests visually analyzing the vectors, while hints 2 and 3 suggest using Gaussian elimination to determine independence and hint 4 mentions the rank-nullity theorem as a possible approach.
  • #1
TomSavage
4
0
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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  • #2
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.
 

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

1. What are vectors S and T in the column of (ABC)?

Vectors S and T refer to two specific mathematical quantities that are represented by columns in the matrix (ABC). They are often used in linear algebra and represent both magnitude and direction.

2. How are vectors S and T related to the matrix (ABC)?

Vectors S and T are columns in the matrix (ABC), meaning that they are part of the larger set of data represented by the matrix. They can be used to perform calculations and transformations on the matrix as a whole.

3. Can vectors S and T be used interchangeably in the matrix (ABC)?

No, vectors S and T are distinct quantities and cannot be used interchangeably in the matrix (ABC). Each vector has its own unique properties and cannot be substituted for the other.

4. How are vectors S and T calculated in the matrix (ABC)?

Vectors S and T are typically calculated using a combination of mathematical operations, such as addition, subtraction, and multiplication, on the elements of the matrix (ABC). The specific calculations will depend on the context in which the matrix is being used.

5. Why are vectors S and T important in the column of (ABC)?

Vectors S and T play a crucial role in understanding and manipulating the data represented by the matrix (ABC). They allow for complex calculations and transformations to be performed on the matrix, making it a powerful tool in various scientific fields.

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