Linear Algebra Vector Transformations

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Questions and Answers

What does the equation $x_i = l_{ji} x'_j$ represent in the context of vectors?

Transformation of coordinates into a new system. (correct)

The dot product of two vectors.

Interaction of two vectors in the same axis.

The cross product of two vectors.

What implies that a matrix $L$ is 'proper orthogonal'?

All elements of the matrix are positive.

The matrix is symmetric.

The determinant of the matrix is +1. (correct)

All rows are orthogonal to each other.

How is the rotation matrix related to the identity matrix according to the provided content?

$L^2 = I$ indicates the matrix is singular.

$L*L = I$ indicates symmetry.

$L^T L = I$ indicates linear independence. (correct)

$LL^T = I$ indicates orthogonality. (correct)

What is indicated when the determinant of the rotation matrix $L$ is -1?

There is a rotation combined with a reflection. Signup and view all the answers

In the expression $x_i e_i = x'_j e'_j$, what does $e_i$ and $e'_j$ represent?

Basis vectors in different coordinate systems. Signup and view all the answers

What conclusion can be drawn from the equation $l_{ik} l_{kj} = oldsymbol{ au}_{ij}$?

It shows a direct transformation between two tensors. Signup and view all the answers

Which of the following equations shows the relationship between the old and new basis vectors?

$e_i' = l_{ji} e_j$ Signup and view all the answers

What is implied by the equation $det L = ±1$ for the rotation matrix?

It distinguishes types of transformations. Signup and view all the answers

What does the Kronecker delta function, indicated as $oldsymbol{ au}_{ij}$, represent in the context of the provided equations?

It represents the identity matrix. Signup and view all the answers

What defines a tensor of order one in the context of this document?

It refers to a one-dimensional array. Signup and view all the answers

What is the result of contracting the dyadic product of two vectors?

It becomes a scalar quantity. Signup and view all the answers

What does the trace of a tensor represent?

The diagonal components of a tensor summed together. Signup and view all the answers

Which statement correctly defines a symmetric tensor?

It remains unchanged when indices are swapped. Signup and view all the answers

How is an isotropic tensor characterized?

By remaining invariant under rotations of the basis vectors. Signup and view all the answers

When contracting a tensor of order four (𝐶𝑇(4)), what is the resulting order?

It reduces to 𝐶𝑇(2). Signup and view all the answers

What indicates that a tensor is skew-symmetric?

Its components satisfy 𝑇𝑖𝑗 = −𝑇𝑗𝑖. Signup and view all the answers

What is the condition under which a Cartesian tensor of order two cannot be represented as a dyadic product?

If at least two indices are equal Signup and view all the answers

What does the notation 𝑇𝑖𝑗𝑘𝑘 represent when contracting indices?

A symmetric tensor. Signup and view all the answers

When considering a cartesian tensor of order two (𝐶𝑇(2)), how many independent variables does a skew-symmetric tensor possess?

3 independent variables. Signup and view all the answers

If tensors T and S are of orders n and m respectively, what is the resultant order of their product U = S T?

n + m Signup and view all the answers

Which operation on tensors results in the decrease of the tensor's order?

Contraction of tensors Signup and view all the answers

What characterizes the transformation of components when a basis vector rotates for isotropic tensors?

They remain constant regardless of the rotation. Signup and view all the answers

What does the equation $T' = L T L^T$ represent?

Representation of T in a different basis Signup and view all the answers

In a tensor represented by $T_{ij}$, which of the following represents the dyadic product?

$u_i v_j$ for specific vectors u and v Signup and view all the answers

What will be the form of a Cartesian tensor of order two if it is expressed in terms of basis vectors?

$T = T_{ij} (e_i ensor e_j)$ Signup and view all the answers

What happens when the components of a tensor $T$ are contracted with respect to two indices?

The resulting tensor's order decreases by 2 Signup and view all the answers

Which of the following statements about tensor equations is true?

They are independent of any basis Signup and view all the answers

In the transformation law for tensor $T$, what symbol represents the transformed tensor?

$T'$ Signup and view all the answers

What is the result of the product of two axial tensors?

A proper tensor Signup and view all the answers

What happens when you contract an axial tensor of order n?

It produces an axial tensor of order (n - 2) Signup and view all the answers

Which of the following statements about addition of axial tensors is true?

The sum is an axial tensor of the same order Signup and view all the answers

What is the characteristic equation for a CT(2) tensor?

t^3 - I1t^2 + I2t - I3 = 0 Signup and view all the answers

Which of the following correctly represents the principal invariant I1 for a tensor T?

I1 = T11 + T22 + T33 Signup and view all the answers

According to the transformation law of axial tensors, what is the role of Delta (∆)?

It is a scalar factor that can influence the transformation Signup and view all the answers

What type of vector is the curl u represented as in the given equations?

An axial vector Signup and view all the answers

If T_ijk is an axial tensor of order 3, what would be the result of its contraction with another axial tensor of order 2?

An axial tensor of order 1 Signup and view all the answers

How is the position vector expressed for the new axes in relation to the unit vectors?

$x = (x'_1, x'_2, x'_3) = x_j e'_j$ Signup and view all the answers

What does the symbol $ℓ_{ij}$ represent in the transformation law?

The dot product of the old and new unit vectors Signup and view all the answers

What happens to the position vector of point P when the axes are rotated?

It transforms according to the new axes. Signup and view all the answers

Which of the following is true about the relationship between the old and new unit vectors?

$e_i' = ℓ_{ij} e_j$ Signup and view all the answers

Which expression correctly represents the transformation of unit vectors?

$e'i = ℓ{ij} e_j$ Signup and view all the answers

What is the significance of the formula derived for $oldsymbol{x'_i}$ in the context of vector transformations?

It provides a method to transfer vectors when rotated. Signup and view all the answers

In the context of the transformation law, how is the vector $u$ represented?

$u = (u_j e_j) e'_i$ Signup and view all the answers

What does the equation $x_i' = (e_j ⋅ e'_i) x_j$ signify?

It expresses the position vector in a new coordinate system. Signup and view all the answers

Which statement accurately describes how the axes are oriented after rotation?

They are still perpendicular but now reflect different unit vectors. Signup and view all the answers

Which property holds true for the transformation of unit vectors based on the derived equations?

The transformation preserves the orthogonality of the unit vectors. Signup and view all the answers

Study Notes

Tensor Transformation and Rotation

Coordinates in a system with perpendicular axes are represented using unit vectors ( e_1, e_2, e_3 ).
Upon rotation to new unit vectors ( e'_1, e'_2, e'_3 ), the position vector can be expressed in both old and new axes.
The transformation of vectors under rotation can be expressed as ( x'i = \ell{ij} x_j ), where ( \ell_{ij} ) are transformation coefficients.
For unit vectors, ( e'i = \ell{ij} e_j ).

Rotation Matrix Properties

The rotation matrix ( L = [\ell_{ij}] ) captures the relations between old and new coordinate systems.
The orthogonality condition ( L L^T = I ) indicates the preservation of vector lengths.
Determinant of the rotation matrix can be ( +1 ) (proper rotation) or ( -1 ) (rotation plus reflection) indicating symmetry or inversion under rotation.

Tensor Representation

Cartesian tensors of order two cannot be expressed as dyadic products; instead, they can be represented through summation over basis vectors.
A second-order tensor can transform according to the law: ( T'{pq} = \ell{pi} \ell_{qj} T_{ij} ).
Resulting first-order tensors like traces (e.g., ( T_{qq} )) serve as invariants or scalars.

Tensor Operations

Scalar multiplication and addition of tensors preserve the order of tensors, forming a Cartesian tensor of the same order.
Contraction reduces the order of tensors; contracting indices gives another tensor of lower order.
For example, contracting a tensor of order ( n ) results in a tensor of order ( n-2 ).

Symmetric and Anti-Symmetric Tensors

A tensor is symmetric if ( T_{ij} = T_{ji} ) and anti-symmetric if ( T_{ij} = -T_{ji} ).
Every tensor can be decomposed into its symmetric and skew-symmetric components.

Isotropic Tensors

Tensors that remain unchanged under any arbitrary rotation are termed isotropic.
The transformation law for an axial tensor involves a sign change (±1) and is defined in higher dimensions.

Cayley-Hamilton Theorem

Every second-order Cartesian tensor satisfies its own characteristic equation, providing a relationship between its eigenvalues and invariants.
The cubic polynomial derived from the determinant of the matrix form of the tensor plays a crucial role in spectral analysis.

Important Notes

Determining trace and scalar characteristics of tensors is critical for understanding their physical quantities and properties.
Properly identifying symmetric and anti-symmetric parts of tensors aids in applications such as mechanics and material science.
The transformational laws allow for the analysis and derivation of tensor properties across different coordinate systems, making them foundational in fields such as physics and engineering.

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Description

This quiz focuses on the concepts of vector transformations in linear algebra, specifically covering the transformation laws and the relationships between unit vectors in a coordinate system. Test your understanding of cosine relationships and vector notation.