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Matrix multiplication is a key operation in linear algebra that combines two matrices to produce a new matrix. It's important to understand how this operation works because many concepts in linear algebra, like transformations and solving systems of equations, rely on it.
When multiplying two matrices, each element of the resulting matrix is obtained by taking the dot product of rows from the first matrix and columns from the second matrix. To perform matrix multiplication:

• Ensure the number of columns in the first matrix equals the number of rows in the second matrix.
• Calculate the dot product for each row and column combination.
• Place the result in the corresponding position in the new matrix.

In the context of this exercise, we used matrix multiplication to factor out the matrix \(C\) from a linear combination of vectors. This allowed us to transition from a complex vector equation to a simpler form where the properties of matrix \(C\) could be utilized.

An invertible matrix, also known as a nonsingular or nondegenerate matrix, is a square matrix that has an inverse. When a matrix \(C\) is invertible, it means there exists another matrix, \(C^{-1},\) such that the multiplication of \(C\) and \(C^{-1}\) gives the identity matrix.
In linear algebra, invertible matrices are crucial because they ensure unique solutions exist for systems of linear equations. Some key properties of invertible matrices include:

• An invertible matrix transforms a linear system such that there is exactly one solution.
• If \(C\) is invertible, the equation \(C\mathbf{x} = \mathbf{0}\) implies that \(\mathbf{x} = \mathbf{0}\).
• The inverse of an invertible matrix is unique.

In this exercise, because \(C\) is invertible, this property helps us conclude that if the transformation of vectors results in a zero vector, the original vector coefficients must be zero, proving the independence of vectors \(\mathbf{Cv}_1, \mathbf{Cv}_2, \ldots, \mathbf{Cv}_k\).

Column vectors are vertical arrays of numbers. In the context of linear algebra, they are used to represent coordinates or directions in a space. For instance, a column vector in \(\mathbb{R}^n\) has \(n\) elements and can represent a point in an \(n\)-dimensional space.
When dealing with matrices, each column consists of entries from particular linear combinations, effectively defining a direction in the vector space. Here are some points about column vectors:

• They are typically expressed as single columns (instead of rows).
• When used in matrices, each column effectively amalgamates multiple inputs to produce outputs through linear transformations.
• Linear independence among column vectors means that no vector in the set can be written as a linear combination of the others.

In this exercise, column vectors \(\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\) are given to be independent. This concept is utilized in proving that the transformed vectors \(\mathbf{Cv}_1, \mathbf{Cv}_2, \ldots, \mathbf{Cv}_k\) also maintain independence, underpinning the strength of linear transformations applied to independent sets of vectors.