True or False: Given an array of features x and an array of corresponding coefficients c, you can compute a predicted value by doing x @ c

Question

BlackTom AI · Accepted Answer

Matrix multiplication (the dot product) x @ c produces the linear combination of features with their coefficients, which is exactly the predicted value for a linear model.

1) Relevant knowledge points:
- The @ operator in NumPy performs matrix multiplication (or dot product for 1-D arrays).
- A predicted value in a linear model is the sum of each feature times its coefficient: sum_i x_i * c_i.
- Shape compatibility matters: for a single sample x as a 1-D array and c as a 1-D coefficients array of the same length, x @ c yields a scalar. For many samples, X (shape (n_samples, n_features)) @ c (shape (n_features,)) yields a prediction vector of length n_samples.

2) Step-by-step reasoning:
- Represent features for one sample as x = [x1, x2, x3] and coefficients as c = [c1, c2, c3].
- The @ operator computes x @ c = x1*c1 + x2*c2 + x3*c3.
- For a batch: X = [[x11,x12,x13],[x21,x22,x23]] shape (2,3) and c = [c1,c2,c3] shape (3,), then X @ c = [x11*c1+x12*c2+x13*c3, x21*c1+x22*c2+x23*c3] shape (2,).

3) Calculations / intermediate steps (single sample):
x = [2, 5, -1], c = [0.5, 1.0, 2.0]
x @ c = 2*0.5 + 5*1.0 + (-1)*2.0 = 1.0 + 5.0 - 2.0 = 4.0

4) Why the correct answer is correct:
This operation implements the linear model formula y_hat = Xc. The @ operator is the direct, idiomatic way in NumPy/Python to compute these dot products or matrix-vector products, returning the predicted scalar or vector as appropriate when shapes align.

5) Why the other option is incorrect:
Answer "False" would imply that x @ c cannot compute predictions; that is incorrect because when x and c have compatible shapes (or when X is a 2-D feature matrix and c a coefficient vector) x @ c exactly yields predicted values. The only caveats are shape mismatches or omission of an intercept term if required — these are usage issues, not a failure of the operator.

6) Conclusion:
Therefore the statement is True. This matches the provided correct answer: True.

True or False: Given an array of features x and an array of corresponding coefficients c, you can compute a predicted value by doing x @ c判断题

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