Complete this scalar multiplication: The solution will be:

Question

BlackTom AI · Accepted Answer

Scalar multiplication multiplies each component of the vector by the scalar. For a scalar c and vector v = [v1, v2, v3]^T, c v = [c·v1, c·v2, c·v3]^T.

1) Relevant concepts
- Definition of scalar multiplication in R^n: multiply each entry of the vector by the scalar.
- No change of order or sign conventions beyond ordinary arithmetic.

2) Step-by-step reasoning
We are given the scalar -5 and the column vector [1, 0, 9]^T. Apply the scalar to each component individually.

3) Calculations (all intermediate steps)
- First component: -5 × 1 = -5
- Second component: -5 × 0 = 0
- Third component: -5 × 9 = -45

So performing the multiplication entrywise yields the resulting vector [-5, 0, -45]^T.

4) Why the correct answer is correct
The displayed solution vector in the question is exactly [-5, 0, -45]^T, which matches the entrywise products computed above. This follows directly from the definition of scalar multiplication: the scalar -5 multiplies each element of the original vector.

5) Why the other option is incorrect
The only other choice is False. That would mean the given resulting vector is not equal to -5 times [1,0,9]^T. Since the entrywise multiplication produces exactly [-5, 0, -45]^T, the False option contradicts those correct arithmetic results and is therefore incorrect.

6) Conclusion
The proposed solution vector [-5, 0, -45]^T is correct, so the correct selection is True, confirming the provided answer.

Complete this scalar multiplication: The solution will be:True/False

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