True/False question
Suppose that A is a 2 x 3 matrix such that A[1, 1, 1]^t = [2, 3]^t = A[2, 3, 4]^t. Then [1, 2, 3]^t belongs to the nullspace of A.
I have no idea how to start this question/what it means.
First, be sure you understand a matrix null space. Google is your friend.
Te null space is all vectors x such that
xA = 0
I'm having a hard time understanding
A[1, 1, 1]^t = [2, 3]^t = A[2, 3, 4]^t
How can they all be equal?
To determine whether [1, 2, 3] belongs to the nullspace of matrix A, we need to understand what the nullspace is and how to find it.
The nullspace of a matrix consists of all vectors that, when multiplied by the matrix, result in the zero vector. In other words, for a matrix A, the nullspace is the set of all solutions to the equation A*x = 0, where 'x' is a vector.
Now let's interpret the given information:
A is a 2 x 3 matrix, meaning it has 2 rows and 3 columns. We are given that A[1, 1, 1]^t = [2, 3]^t and A[2, 3, 4]^t = [2, 3]^t.
The superscript 't' denotes the transpose of a vector or matrix. Transposing a vector or matrix means interchanging its rows and columns.
Let's break down the given equations:
1. A[1, 1, 1]^t = [2, 3]^t:
This equation tells us that when we multiply matrix A by the column vector [1, 1, 1]^t, we obtain the column vector [2, 3]^t.
2. A[2, 3, 4]^t = [2, 3]^t:
This equation states that when we multiply matrix A by the column vector [2, 3, 4]^t, we get the same column vector [2, 3]^t as in equation 1.
Now, to determine whether [1, 2, 3]^t belongs to the nullspace of A, we need to solve the equation A*x = 0, where 'x' is the unknown vector we seek. In other words, we want to find 'x' such that the product of matrix A and vector x is the zero vector.
However, since A is not explicitly given nor is the nullspace condition stated, we cannot definitively answer the true/false question without more information.
To solve this further, we would need to know the specific values of matrix A or additional conditions.