For Exercises 25–34, use inductive reasoning to find a pattern for the answers. Then use the pattern to guess the result of the final calculation, and perform the operation to see if your answer is correct . 25. 12 , 345 , 679 × 9 = 111 , 111 , 111 12 , 345 , 679 × 18 = 222 , 222 , 222 12 , 345 , 679 × 27 = 333 , 333 , 333 ⋮ 12 , 345 , 679 × 72 = ?
For Exercises 25–34, use inductive reasoning to find a pattern for the answers. Then use the pattern to guess the result of the final calculation, and perform the operation to see if your answer is correct . 25. 12 , 345 , 679 × 9 = 111 , 111 , 111 12 , 345 , 679 × 18 = 222 , 222 , 222 12 , 345 , 679 × 27 = 333 , 333 , 333 ⋮ 12 , 345 , 679 × 72 = ?
Solution Summary: The author explains that inductive reasoning is the process of reasoning to a general conclusion through observations of specific cases.
For Exercises 25–34, use inductive reasoning to find a pattern for the answers. Then use the pattern to guess the result of the final calculation, and perform the operation to see if your answer is correct.
19. Block Matrices
• Prove that the determinant of a block matrix:
A B
0 D
.
is det(A) · det (D), where A and D are square matrices.
•
Show how block matrices are used in solving large-scale linear systems.
6. Norms and Metrics
•
Show that the function || ||
norm on Rn.
=
√xT Ax, where A is a positive definite matrix, defines a
.
Prove that the matrix norm induced by the vector L²-norm satisfies ||A||2
✓ max (ATA), where Amax is the largest eigenvalue.
2. Linear Transformations
•
•
Let T: R3 R³ be a linear transformation such that T(x, y, z) = (x + y, y + z, z +
→
x). Find the matrix representation of T with respect to the standard basis.
Prove that a linear transformation T : VV is invertible if and only if it is bijective.
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