[M] Determine whether 𝐰 is in the column space of A, the null space of A, or both, where 𝐰=[

Name: Problem 38, Chapter 4: Linear Algebra and Its Applications
Uploaded: 2019-08-11T02:36:36-08:00
Duration: 1 min 33 s
Channel: Doruk Isik
Description: [M] Determine whether 𝐰 is in the column space of A, the null space of A, or both, where 𝐰=[ 1 2 1 0 ], A=[ -8 5 -2 0 -5 2 1 -2 10 -8 6 -3 3 -2 1 0 ]

step 1 In this problem we're given an vector W and the matrix A and then we're asked to determine wheth

Question

[M] Determine whether $\mathbf{w}$ is in the column space of $A,$ the null space of $A,$ or both, where $$ \mathbf{w}=\left[\begin{array}{l}{1} \\ {2} \\ {1} \\ {0}\end{array}\right], \quad A=\left[\begin{array}{rrrr}{-8} & {5} & {-2} & {0} \\ {-5} & {2} & {1} & {-2} \\ {10} & {-8} & {6} & {-3} \\ {3} & {-2} & {1} & {0}\end{array}\right] $$

   [M] Determine whether $\mathbf{w}$ is in the column space of $A,$ the null space of $A,$ or both, where 
$$
\mathbf{w}=\left[\begin{array}{l}{1} \\ {2} \\ {1} \\ {0}\end{array}\right], \quad A=\left[\begin{array}{rrrr}{-8} & {5} & {-2} & {0} \\ {-5} & {2} & {1} & {-2} \\ {10} & {-8} & {6} & {-3} \\ {3} & {-2} & {1} & {0}\end{array}\right]
$$

Linear Algebra and Its Applications

David C. Lay, Steven… 5th Edition

Chapter 4, Problem 38

Instant Answer

Solved by Expert Doruk Isik

08/11/2019

Step 1

To do this, we need to solve the system $A\mathbf{x}=\mathbf{w}$, where $\mathbf{x}$ is a vector of variables. We can solve this system by augmenting the matrix $A$ with the vector $\mathbf{w}$ and then reducing to row-echelon form. Show more…

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[M] Determine whether $\mathbf{w}$ is in the column space of $A,$ the null space of $A,$ or both, where $$ \mathbf{w}=\left[\begin{array}{l}{1} \\ {2} \\ {1} \\ {0}\end{array}\right], \quad A=\left[\begin{array}{rrrr}{-8} & {5} & {-2} & {0} \\ {-5} & {2} & {1} & {-2} \\ {10} & {-8} & {6} & {-3} \\ {3} & {-2} & {1} & {0}\end{array}\right] $$

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Key Concepts

Column Space

The column space of a matrix is the set of all possible vectors that can be formed as linear combinations of the matrix’s columns. It represents the range of the linear transformation defined by the matrix. In the context of solving systems, if a vector can be produced by multiplying the matrix by some vector, then that vector is part of the column space.

Null Space

The null space (or kernel) of a matrix consists of all vectors that, when multiplied by the matrix, result in the zero vector. It represents the set of solutions to the homogeneous equation Ax = 0. Understanding the null space is crucial for determining the dependencies among the columns of the matrix and for analyzing the behavior of the corresponding linear transformation.

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