Understanding the Properties of Matrix Multiplication for Basis Determination | Debunking a False Statement

If A is a 7 × 7 matrix and the columns of A^10 form a basis for R7, then the columns for A itself also form a basis for R7. (T/F)

False

False.

In order to determine the validity of this statement, we must understand the properties of matrix multiplication and the concept of a basis.

First, let’s understand the concept of a basis. In linear algebra, a basis for a vector space is a set of vectors that are linearly independent and that span the entire space. In other words, if a set of vectors forms a basis for a vector space, we can express any other vector in that space as a unique linear combination of those basis vectors.

Now, let’s consider the given statement. It states that if the columns of A^10 (the 10th power of matrix A) form a basis for R7 (the vector space of column vectors of size 7), then the columns of A itself will also form a basis for R7.

Matrix multiplication has the property that multiplying two matrices together creates a new matrix with a column space that is a subset of the column space of the multiplied matrices. In other words, the columns of the product matrix are linear combinations of the columns of the original matrices.

Considering this property, if the columns of A^10 form a basis for R7, it means that any column vector in R7 can be expressed as a linear combination of the columns of A^10. However, it does not necessarily mean that any column vector in R7 can be expressed as a linear combination of the columns of A itself.

To illustrate this, let’s consider a simple example. Suppose we have a 2×2 matrix A with columns [1 0] and [0 1]. The columns of A^10 would be [1 0] and [0 1]. Clearly, the columns of A form a basis for R2 as they are linearly independent and span the space. However, in this example, the columns of A^10 also form a basis for R2, but this does not mean that the columns of A itself do.

Therefore, we can conclude that the statement is false.

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