A data scientist is working with a dataset containing 10,000 samples and 784 features, represented as a data matrix D with dimensions 10,000 x 784. The goal is to apply a linear transformation to this dataset to reduce the number of features to 64. This transformation is achieved by right-multiplying the data matrix D by a transformation matrix T, resulting in a new matrix D_prime. What must be the dimensions of the transformation matrix T for this operation to be valid, and what will be the dimensions of the resulting matrix D_prime?
Matrix T must be 784 x 64, and D_prime will be 10,000 x 64.
Matrix T must be 10,000 x 64, and D_prime will be 10,000 x 64.
Matrix T must be 64 x 784, and D_prime will be 10,000 x 784.
Matrix T must be 784 x 784, and D_prime will be 10,000 x 784.
The correct answer specifies that the transformation matrix T must have dimensions 784 x 64, and the resulting matrix D_prime will have dimensions 10,000 x 64.
Explanation:
The fundamental rule for matrix multiplication states that for the product of two matrices, A * B, to be defined, the number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B). In this scenario, the operation is D * T = D_prime.
The data matrix D has dimensions m x n, where m = 10,000 (samples/rows) and n = 784 (features/columns).
The transformation matrix T must have dimensions n x k, where n is the number of rows and k is the number of columns.
For the multiplication D * T to be valid, the number of columns in D (784) must equal the number of rows in T. Therefore, T must have 784 rows.
The goal is to reduce the feature dimension to 64, which means the resulting matrix D_prime must have 64 columns. The number of columns in the resulting matrix is determined by the number of columns in the second matrix (T). Therefore, T must have 64 columns.
Combining these requirements, the dimensions of T must be 784 x 64.
The resulting matrix, D_prime, will have the number of rows from the first matrix (D) and the number of columns from the second matrix (T). Thus, the dimensions of D_prime will be 10,000 x 64.
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Why does the transformation matrix T need to have dimensions 784 x 64 in this scenario?
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How is dimensionality reduction achieved through matrix multiplication?