Questions and Answers
The statement $||v|| = \sqrt{v \cdot v}$ is true.
True
For any scalar $c$, it is true that $u \cdot (cv) = c(u \cdot v)$.
True
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 $Nul A$.
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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 $W^\perp$.
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$u \cdot v - v \cdot u = 0$.
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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 $W^\perp$.
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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 $A$.
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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.
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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.
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A matrix with orthonormal columns is an orthogonal matrix.
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The distance from $y$ to line $L$ can be determined from $||y - \hat{y}||$.
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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.
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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.
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The orthogonal projection of $y$ onto $v$ is the same as the orthogonal projection of $y$ onto $cv$ whenever $c \neq 0$.
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An orthogonal matrix is invertible.
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Why is UV invertible?
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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?
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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 ).
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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!
