Podcast
Questions and Answers
The statement $||v|| = \sqrt{v \cdot v}$ is true.
The statement $||v|| = \sqrt{v \cdot v}$ is true.
True
For any scalar $c$, it is true that $u \cdot (cv) = c(u \cdot v)$.
For any scalar $c$, it is true that $u \cdot (cv) = c(u \cdot v)$.
True
If the distance from $u$ to $v$ equals the distance from $u$ to $-v$, then $u$ and $v$ are orthogonal.
If the distance from $u$ to $v$ equals the distance from $u$ to $-v$, then $u$ and $v$ are orthogonal.
True
For a square matrix $A$, vectors in $Col A$ are orthogonal to vectors in $Nul A$.
For a square matrix $A$, vectors in $Col A$ are orthogonal to vectors in $Nul A$.
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If vectors $v_1, ..., v_p$ span a subspace $W$ and $x$ is orthogonal to each $v_j$ for $j=1,...,p$, then $x$ is in $W^\perp$.
If vectors $v_1, ..., v_p$ span a subspace $W$ and $x$ is orthogonal to each $v_j$ for $j=1,...,p$, then $x$ is in $W^\perp$.
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$u \cdot v - v \cdot u = 0$.
$u \cdot v - v \cdot u = 0$.
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For any scalar $c$, $||cv|| = c||v||$.
For any scalar $c$, $||cv|| = c||v||$.
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If $x$ is orthogonal to every vector in a subspace $W$, then $x$ is in $W^\perp$.
If $x$ is orthogonal to every vector in a subspace $W$, then $x$ is in $W^\perp$.
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If $||u||^2 + ||v||^2 = ||u + v||^2$, then $u$ and $v$ are orthogonal.
If $||u||^2 + ||v||^2 = ||u + v||^2$, then $u$ and $v$ are orthogonal.
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For an $m \times n$ matrix $A$, vectors in the null space of $A$ are orthogonal to vectors in the row space of $A$.
For an $m \times n$ matrix $A$, vectors in the null space of $A$ are orthogonal to vectors in the row space of $A$.
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Not every linearly independent set in $,R^n$ is an orthogonal set.
Not every linearly independent set in $,R^n$ is an orthogonal set.
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If $y$ is a linear combination of nonzero vectors from an orthogonal set, then the weights can be computed without row operations on a matrix.
If $y$ is a linear combination of nonzero vectors from an orthogonal set, then the weights can be computed without row operations on a matrix.
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If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal.
If the vectors in an orthogonal set of nonzero vectors are normalized, then some of the new vectors may not be orthogonal.
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A matrix with orthonormal columns is an orthogonal matrix.
A matrix with orthonormal columns is an orthogonal matrix.
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The distance from $y$ to line $L$ can be determined from $||y - \hat{y}||$.
The distance from $y$ to line $L$ can be determined from $||y - \hat{y}||$.
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Not every orthogonal set in $,R^n$ is linearly independent.
Not every orthogonal set in $,R^n$ is linearly independent.
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If a set $S = {u_1, ..., u_p}$ has the property that $u_i \cdot u_j = 0$ whenever $i \neq j$, then $S$ is an orthonormal set.
If a set $S = {u_1, ..., u_p}$ has the property that $u_i \cdot u_j = 0$ whenever $i \neq j$, then $S$ is an orthonormal set.
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If the columns of an $m \times n$ matrix $A$ are orthonormal, then the linear mapping $x \mapsto Ax$ preserves lengths.
If the columns of an $m \times n$ matrix $A$ are orthonormal, then the linear mapping $x \mapsto Ax$ preserves lengths.
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The orthogonal projection of $y$ onto $v$ is the same as the orthogonal projection of $y$ onto $cv$ whenever $c \neq 0$.
The orthogonal projection of $y$ onto $v$ is the same as the orthogonal projection of $y$ onto $cv$ whenever $c \neq 0$.
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An orthogonal matrix is invertible.
An orthogonal matrix is invertible.
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Why is UV invertible?
Why is UV invertible?
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For any two $n \times n$ invertible matrices $U$ and $V$, the inverse of $UV$ is _____.
For any two $n \times n$ invertible matrices $U$ and $V$, the inverse of $UV$ is _____.
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How can this inverse be expressed using transposes?
How can this inverse be expressed using transposes?
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To show that $(UV)^{-1} = (UV)^T$, apply the property that states $(UV)^T = _____.
To show that $(UV)^{-1} = (UV)^T$, apply the property that states $(UV)^T = _____.
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Study Notes
Properties of Vectors and Inner Products
- The length of a vector ( v ) is expressed as ( |v| = \sqrt{v \cdot v} ).
- The inner product is commutative, leading to ( u \cdot v - v \cdot u = 0 ).
- For any scalar ( c ), ( |cv| = |c||v| ), showing that if ( c ) is negative, the length remains positive.
Orthogonality in Linear Algebra
- Vectors ( u ) and ( v ) are orthogonal if ( u \cdot v = 0 ). This is equivalent to equal distances from ( u ) to ( v ) and ( -v ).
- A vector ( x ) is in the orthogonal complement ( W^\perp ) of a subspace ( W ) if it is orthogonal to every vector in ( W ).
- By the Pythagorean theorem, two vectors ( u ) and ( v ) are orthogonal if ( |u + v|^2 = |u|^2 + |v|^2 ).
Null Spaces and Row Spaces
- For a matrix ( A ), vectors in the null space are orthogonal to vectors in the row space, defined by ( (\text{Row } A)^\perp = \text{Nul } A ).
- Not every linearly independent set in ( \mathbb{R}^n ) is orthogonal, evidenced by examples of independent but non-orthogonal vectors.
Orthogonal Sets and Linear Combinations
- Coefficients in a linear combination of an orthogonal set can be determined without row operations: ( c_j = \frac{y \cdot u_j}{u_j \cdot u_j} ).
- Orthogonal sets of nonzero vectors are always linearly independent, while not every orthogonal set guarantees linear independence.
Essential Matrix Properties
- A matrix with orthonormal columns is not necessarily orthogonal unless it is square.
- The orthogonal projection of ( y ) onto a line ( L ) does not directly yield the distance from ( y ) to ( L ); rather, it is ( |y - \hat{y}| ) that provides this distance.
- If ( U ) and ( V ) are orthogonal matrices, then the product ( UV ) is also invertible, and ( (UV)^{-1} = V^{-1}U^{-1} ).
Inverses and Transpose Relationships
- For orthogonal matrices, the inverse is given by the transpose: ( U^{-1} = U^T ).
- The inverse of the product of two matrices relates to the transposes of the individual matrices as ( (UV)^{-1} = V^T U^T ).
Studying That Suits You
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Test your knowledge on linear algebra concepts with these flashcards. Each card features a term or statement followed by an explanation of its validity within the context of inner products and vector norms. Perfect for review or self-study!
