Two Geometric Interpretations of Multiplying a Matrix and a Vector

In this video, we'll explore two geometric interpretations of multiplying a matrix and a vector. The first interpretation involves transforming a vector into another vector using the matrix. The second interpretation combines the columns of the matrix to produce a new vector.

For the first interpretation, consider a coordinate system with a vector (1, 2). When we multiply this vector by the matrix [1, 2, 3; 4, 5, 6], we get a new vector (5, 11). This interpretation shows how the matrix acts on the vector to produce a transformation.

In the second interpretation, we view the multiplication as combining the columns of the matrix. The vector (1, 2) is split into components (1 and 2). We multiply each component with the corresponding column of the matrix and sum them up. This results in the same vector (5, 11) as before.

Exercise 1 Consider a matrix [2, 1; 3, 2] and a vector (1, 3). Calculate the result of multiplying the matrix by the vector using both interpretations.

Solution 1
Interpretation 1 [2, 1; 3, 2] * (1, 3) = (8, 11)
Interpretation 2 [2, 1; 3, 2] * (1, 3) = (8, 11)

Exercise 2 Let's try a different example. Multiply the matrix [1, 2; 2, 3] by the vector (3, 1) using both interpretations.

Solution 2
Interpretation 1 [1, 2; 2, 3] * (3, 1) = (5, 7)
Interpretation 2 [1, 2; 2, 3] * (3, 1) = (5, 7)

Exercise 3 Explore the geometric interpretations of matrix-vector multiplication for a matrix and vector of your choice.

Solution 3
Interpretation 1 Perform the dot product of each row with the vector's components.
Interpretation 2 Split the vector into components and combine with the columns of the matrix.

Remember, these interpretations provide different ways of understanding matrix-vector multiplication, but they yield the same result.

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