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Questions and Answers
If x and y are orthogonal, then x × y = 1
If x and y are orthogonal, then x × y = 1
False (B)
If Ax = 0 for some x ≠ 0, then A is invertible
If Ax = 0 for some x ≠ 0, then A is invertible
False (B)
If A is a 5×5 matrix and det(A) = 4, then det(2A) = 20
If A is a 5×5 matrix and det(A) = 4, then det(2A) = 20
False (B)
If Ax = λx, then A²x = λ²x
If Ax = λx, then A²x = λ²x
If A has 0 as an eigenvalue, then A is not invertible
If A has 0 as an eigenvalue, then A is not invertible
The determinant of an elementary matrix is always nonzero
The determinant of an elementary matrix is always nonzero
If A is symmetric, then A⁻¹ is symmetric
If A is symmetric, then A⁻¹ is symmetric
If x and y are two vectors in the x-y plane of R³, then 4x × y is a vector parallel to the z-axis
If x and y are two vectors in the x-y plane of R³, then 4x × y is a vector parallel to the z-axis
If A and B are both symmetric n×n matrices, then AB is also symmetric
If A and B are both symmetric n×n matrices, then AB is also symmetric
If Ax = 0, then A(2x) = 0
If Ax = 0, then A(2x) = 0
For any x and y in R³, x × y = y × x
For any x and y in R³, x × y = y × x
For any x and y in R³, x v = y x
For any x and y in R³, x v = y x
If x y = 0 and y z = 0, then x z = 0
If x y = 0 and y z = 0, then x z = 0
Det(A + B) = det(A) + det(B)
Det(A + B) = det(A) + det(B)
Det(AB) = det(A)det(B)
Det(AB) = det(A)det(B)
Det(rA) = rdet(A)
Det(rA) = rdet(A)
If AB = 0 and A ≠ 0, then B = 0
If AB = 0 and A ≠ 0, then B = 0
If T: R³ → R³ is a linear transformation, then T²(x) = T(x²)
If T: R³ → R³ is a linear transformation, then T²(x) = T(x²)
Suppose λ is an eigenvalue of the matrix A with x as its corresponding eigenvector. Then A²(x) = λ²x
Suppose λ is an eigenvalue of the matrix A with x as its corresponding eigenvector. Then A²(x) = λ²x
If x is a solution to Ax = 0, then so is 2x
If x is a solution to Ax = 0, then so is 2x
For any x and y in R³, x × y = -y × x
For any x and y in R³, x × y = -y × x
For any x and y in R³, x y = -y x
For any x and y in R³, x y = -y x
If x y = 0 and y z = 0, then (x + z) y = 0
If x y = 0 and y z = 0, then (x + z) y = 0
The determinant of the matrix |0 0 3| |0 2 5| = 6 |1 4 6|
The determinant of the matrix |0 0 3| |0 2 5| = 6 |1 4 6|
If A ≠ 0 and B ≠ 0, then AB ≠ 0
If A ≠ 0 and B ≠ 0, then AB ≠ 0
If T: R³ → R³, then T(x) T(y) = T(x × y)
If T: R³ → R³, then T(x) T(y) = T(x × y)
The image vector of a linear operator that first rotates the vector (1,1) by 90 degrees then contracts the vector by a factor k = 0.5 is (0.5,0.5).
The image vector of a linear operator that first rotates the vector (1,1) by 90 degrees then contracts the vector by a factor k = 0.5 is (0.5,0.5).
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Study Notes
Orthogonality and Eigenvalues
- Orthogonal vectors do not satisfy x × y = 1; this statement is false.
- A matrix A is not invertible if there exists a non-zero vector x such that Ax=0.
- If det(A)=4 for a 5×5 matrix A, then det(2A) is actually 32, not 20.
Eigenvalue Properties
- If Ax=λx, then applying A again yields A²x=λ²x, confirming the relation holds true.
- A matrix A cannot be invertible if it has an eigenvalue of 0.
Matrix Determinants and Operations
- The determinant of any elementary matrix is always non-zero.
- The inverse of a symmetric matrix A remains symmetric.
- For vectors in the x-y plane, the cross product 4x × y results in a vector aligned with the z-axis.
Symmetric Matrices and Operations
- The product of two symmetric matrices A and B is not guaranteed to be symmetric.
- If Ax=0, then A(2x)=0 is true by scalar multiplication properties.
Vector Operations in R³
- The cross product in R³ is anti-commutative, meaning x × y = -y × x.
- The scalar product of vectors does not adhere to anti-commutativity.
Zero and Non-zero Products
- The expression x · y=0 and y · z=0 does not imply that x · z=0, highlighting the independence of these conditions.
- Determinants do not satisfy the additive property: det(A+B) ≠ det(A) + det(B).
- The multiplicative property holds true under determinants: det(AB) = det(A)det(B).
Linear Transformations and Balances
- A linear transformation T: R³→R³ does not comply with the condition T²(x) = T(x²).
- A true statement in linear solutions is: if x is a solution to Ax = 0, then 2x is also a solution.
Special Cases and Negations
- If x · y=0 and y · z=0, then (x+z) · y=0 holds true due to distributive properties.
- A determinant example with rows containing zeros results in -6, demonstrating incorrect assertion of 6.
Product Negation
- The product of two non-zero matrices A and B can be zero (AB=0) under certain conditions.
- A linear operator does not preserve vector cross products: T(x) · T(y) ≠ T(x × y).
Transformation Misunderstanding
- When performing transformations such as rotations followed by contractions, care must be taken with vector positions. A rotation of vector (1,1) by 90 degrees will result in a coordinate change affecting the sign of the x-component post-transformation.
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