Determine if 𝐰=[ 1 3 -4 ] is in Nul A, where A=[ 3 -5 -3 6 -2 0

step 1 In this example, we have a matrix that's provided and a vector as well, and our goal here is to

Determine if 𝐰=[ 1 3 -4 ] is in Nul A, where...

Determine if $\mathbf{w}=\left[\begin{array}{r}{1} \\ {3} \\ {-4}\end{array}\right]$ is in Nul $A,$ where
$$
A=\left[\begin{array}{rrr}{3} & {-5} & {-3} \\ {6} & {-2} & {0} \\ {-8} & {4} & {1}\end{array}\right]
$$

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Key Concepts

Matrix-Vector Multiplication

Matrix-vector multiplication is a fundamental operation in linear algebra where a matrix is multiplied by a vector to produce another vector. This process is essential for determining if a given vector lies in the null space of a matrix by checking if the product equals the zero vector.

Homogeneous Systems

Homogeneous systems are systems of linear equations where the constant terms are all zero, leading to the form Ax = 0. Solving these systems reveals the null space of the matrix, which is the set of all solutions to the system.

Null Space

The null space of a matrix is the collection of all vectors that, when multiplied by the matrix, yield the zero vector. It represents the solution set to the homogeneous system of linear equations Ax = 0 and is a key concept in understanding the behavior of linear transformations.

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