Determine if the columns of the matrix form a linearly independent set. Justify

aortiH

aortiH

Answered question

2021-09-29

Determine if the columns of the matrix form a linearly independent set. Justify each answer.
[143027514575]

Answer & Explanation

SoosteethicU

SoosteethicU

Skilled2021-09-30Added 102 answers

The given matrix
[143027514575]
The columns of the matrix form a linearly independent set if equation
Ax=0
has only trivial solution
Reduce matrix in reduced echelon form
[143027514575]R1R3R21/2R1R2[45750923232001313]
[45750923232001313]R3+1/4R1R3R3+11/18R2R3[45750923232001313]
[45750923232001313]3R3R3R23/2R3R2[4575092000011]
[4575092000011]R17R3R12/9R2R2[4501201000011]
[4501201000011]R1+5R2R11/4R1R1[100301000011]
Thus. we get equation Ax=0 has not only trivial solution, the columns of the matrix form a linearly dependent set.
Hence, the columns of the matrix form a linearly dependent set
Result: The columns of the matrix form a linearly dependet set

alenahelenash

alenahelenash

Expert2023-06-17Added 556 answers

To determine if the columns of the matrix form a linearly independent set, we need to check if the columns are linearly independent or not.
Let's denote the given matrix as A:
A=[143027514575]
To check for linear independence, we can perform row reduction (Gaussian elimination) on the matrix and see if we obtain any rows of zeros.
Applying row reduction to matrix A:
[143027514575][14300111011195] [1430011100816]
The row-reduced echelon form of the matrix is:
[1430011100816]
Since the row-reduced echelon form has no rows of zeros, we can conclude that the columns of matrix A form a linearly independent set.
Therefore, the columns of the matrix [143027514575] are linearly independent.
star233

star233

Skilled2023-06-17Added 403 answers

Result:
The columns of the matrix form a linearly independent set.
Solution:
Let's denote the given matrix as A:
A=[143027514575]
To check if the columns are linearly independent, we need to solve the equation Ax=0. This can be done by finding the reduced row-echelon form of the augmented matrix [A|0].
Applying row operations, we can perform the following steps:
[143027514575]R2R2+2R1R3R3+4R1[1430011101155]R3R311R2[143001110066]R1R1+16R3R2R2+R3[140101050066]R1R14R2R316R3[1001901050011]
The reduced row-echelon form of the augmented matrix is:
[1001901050011]
From this row-echelon form, we can see that the only solution to the equation Ax=0 is the trivial solution x=[000]. Therefore, the columns of the matrix are linearly independent.

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