Linear Algebra **Determine if the columns of the matrix form a linearly independent set. Justify your answer.**Matrix:\[\begin{bmatrix}-4 & -3 & 0 \\0 & -1 & 4 \\1 & 1 & -4 \\2 & 1 & -8\end{bmatrix}\]**Question:**Select the correct choice below and fill in the answer box within your choice. (Type an integer or simplified fraction for each matrix element.)- **A.** If $ A $ is the given matrix, then the augmented matrix $\begin{bmatrix} \, \, \Box \, \, \end{bmatrix} $ represents the equation $ Ax = 0 $. The reduced echelon form of this matrix indicates that $ Ax = 0 $ has more than one solution. Therefore, the columns of $ A $ do not form a linearly independent set.- **B.** If $ A $ is the given matrix, then the augmented matrix $\begin{bmatrix} \, \, \Box \, \, \end{bmatrix} $ represents the equation $ Ax = 0 $. The reduced echelon form of this matrix indicates that $ Ax = 0 $ has more than one solution. Therefore, the columns of $ A $ form a linearly independent set.- **C.** If $ A $ is the given matrix, then the augmented matrix $\begin{bmatrix} \, \, \Box \, \, \end{bmatrix} $ represents the equation $ Ax = 0 $. The reduced echelon form of this matrix indicates that $ Ax = 0 $ has only the trivial solution. Therefore, the columns of $ A $ do not form a linearly independent set.- **D.** If $ A $ is the given matrix, then the augmented matrix $\begin{bmatrix} \, \, \Box \, \, \end{bmatrix} $ represents the equation $ Ax = 0 $. The reduced echelon form of this matrix indicates that $ Ax = 0 $ has only the trivial solution. Therefore, the columns of $ A $ form a linearly independent set.

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Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Linear Algebra

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

Matrix:

\[
\begin{bmatrix}
-4 & -3 & 0 \\
0 & -1 & 4 \\
1 & 1 & -4 \\
2 & 1 & -8
\end{bmatrix}
\]

**Question:**

Select the correct choice below and fill in the answer box within your choice. (Type an integer or simplified fraction for each matrix element.)

- **A.** If $ A $ is the given matrix, then the augmented matrix $\begin{bmatrix} \, \, \Box \, \, \end{bmatrix} $ represents the equation $ Ax = 0 $. The reduced echelon form of this matrix indicates that $ Ax = 0 $ has more than one solution. Therefore, the columns of $ A $ do not form a linearly independent set.

- **B.** If $ A $ is the given matrix, then the augmented matrix $\begin{bmatrix} \, \, \Box \, \, \end{bmatrix} $ represents the equation $ Ax = 0 $. The reduced echelon form of this matrix indicates that $ Ax = 0 $ has more than one solution. Therefore, the columns of $ A $ form a linearly independent set.

- **C.** If $ A $ is the given matrix, then the augmented matrix $\begin{bmatrix} \, \, \Box \, \, \end{bmatrix} $ represents the equation $ Ax = 0 $. The reduced echelon form of this matrix indicates that $ Ax = 0 $ has only the trivial solution. Therefore, the columns of $ A $ do not form a linearly independent set.

- **D.** If $ A $ is the given matrix, then the augmented matrix $\begin{bmatrix} \, \, \Box \, \, \end{bmatrix} $ represents the equation $ Ax = 0 $. The reduced echelon form of this matrix indicates that $ Ax = 0 $ has only the trivial solution. Therefore, the columns of $ A $ form a linearly independent set.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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