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\(\renewcommand{\R}{\mathbb R }\)
Here you will find the geometric ideas and properties we need from linear algebra.
\(\Leftarrow\) \(\Uparrow\) \(\Rightarrow\)
If \(\mathbf a = (a_1,\ldots, a_n)\) and \(\mathbf b = (b_1,\ldots, b_n)\) are vectors in \(\R^n\), recall the dot product \(\mathbf a\cdot \mathbf b\) is \[ \mathbf a\cdot \mathbf b = a_1 b_1 +\cdots + a_n b_n. \]
The Euclidean norm of \(\mathbf a\) is interpreted as the length of \(\mathbf a\). The Euclidean norm of a difference \(|\mathbf b - \mathbf a|\) is interpreted as the distance between \(\mathbf a\) and \(\mathbf b\), or as the length of the vector \(\mathbf b-\mathbf a\).
Important facts 1. The Triangle Inequality: \[ | \mathbf a + \mathbf b | \le |\mathbf a| + |\mathbf b|. \]
The cosine formula: \[ \mathbf a\cdot \mathbf b = |\mathbf a| |\mathbf b| \cos \theta.\]where \(\theta\) is the angle between vectors \(\mathbf a\) and \(\mathbf b\). Note that the angle between two vectors only makes sense when they are both non-zero vectors, but if one of the vectors is \(\mathbf 0\) the equation becomes \(0=0\).
The Cauchy-Schwarz inequality: \[ |\mathbf a\cdot \mathbf b| \le |\mathbf a|\ |\mathbf b|, \] with equality only if the vectors are linearly dependent.
This definition of orthogonality implies that every vector is orthogonal to the zero vector. For nonzero vectors, orthogonality means the angle \(\theta\) must be \(\pi/2\).
If \(|\mathbf u|=1\), i.e \(\mathbf u\) is a unit vector, and \(\bf v\) is any vector, then \(\bf( u\cdot v) u\) is the projection of \(\bf v\) onto the line generated by \(\bf u\).
If we write \(\bf v = v_1+v_2\), where \(\bf v_1\) is parallel to \(\bf u\) and \(\bf v_2\) is orthogonal to \(\bf u\), then \(\bf v_1= (u\cdot v)u\), and \(\bf u\cdot \bf v=\bf u \cdot \bf v_1 = \pm |\bf u| |\bf v_1|\).
Equivalently, if we form a right triangle whose hypotenuse is \(\bf v\) and whose base lies along the line generated by \(\bf u\) then the base is given by \(\bf (u\cdot v)u\).
In \(\R^n\), we often write \(\mathbf {e_j}\) to denote the unit vector in the \(j\)th coordinate direction. For example, \[ \mathbf {e_1} = (1,0,\ldots, 0), \quad \mathbf {e_2} = (0,1,\ldots, 0), \quad \ldots,\quad \mathbf {e_n} = (0 ,0,\ldots, 1). \]
When \(n=3\) we may write \(\bf i, j, k\) instead of \(\bf e_1, e_2, e_3\), so that \[ \mathbf i = (1,0,0) , \qquad \mathbf j = (0,1,0) , \qquad \mathbf k = (0,0,1) . \]
If we are discussing a vector \(\mathbf a\) in \(\R^n\), then our convention is that \(a_j\) denotes the \(j\)th component, that is, \(a_j = \mathbf a \cdot \mathbf {e_j}\), or \(\mathbf a = (a_1, a_2, \ldots, a_n)\).
In some situations, it is important to distinguish between points and vectors, since these may have very different mathematical or physical interpretations. From this perspective,
a vector normally represents something that is characterized by a direction and a magnitude, such as a force or velocity. A vector is often pictured as an arrow in that direction, with the length indicating the magnitude.
a point normally represents a position. It is often pictured as a point.
But in many situations, the distinction between points and vectors can be ignored with no harm. This is normally the case in MAT237. An element of \(\R^n\) can represent a vector (with a direction and magnitude) or a point (position).
We will however tend to use different letters to indicate what we are thinking of. The letters \(\mathbf u, \mathbf v\) will usually mean “direction-and-magnitude” vectors; while \(\mathbf p, \mathbf x\) will usually mean “point” vectors. We will also often write \({\bf x} = (x,y)\) for a point in \(\R^2,\) and \({\bf x} = (x,y,z)\) for a point in \(\R^3.\)
Thus, a boldface \(\bf x\) denotes an element of \(\R^n\), whereas a non-boldface \(x\) denotes an element of \(\R\). There are different ways to convey the distinction between \(\bf x\) and \(x\) on the blackboard or handwritten notes, for example, \(\vec x\) or \(\underline x\) for \(\mathbf x\). The precise choice does not matter as long as it is reasonable and followed consistently.
Recall that a subspace of Euclidean space \(\R^n\) is a set \(V\) such that if \(\mathbf a, \mathbf b \in V\) then \(c_1\mathbf a + c_2\mathbf b \in V\) for all real numbers \(c_1,c_2\). Note that any subspace must contain the origin, since we can choose \(c_1=c_2=0\).
Suppose that \(A\) is a \(m\times n\) matrix.
If the \(n\) columns of \(A\) are linearly independent, then \(\{ A {\bf x} : {\bf x}\in \R^n \}\) is an \(n\)-dimensional subspace of \(\R^m\). It is the column space or image of \(A\), the subspace consisting of all linear combinations of columns of \(A\). When \(m=3\) and \(n=2\), for example, one can understand geometrically why this space must be a \(2\)-dimensional plane through the origin in \(\R^3\), spanned by the two column vectors of \(A\). The principle is the same in the general case.
If the \(m\) rows of \(A\) are linearly independent, then \(\{ {\bf x}\in \R^n : A {\bf x}= {\bf 0} \}\) is an \((n-m)\)-dimensional subspace of \(\R^n.\) It is the nullspace or kernel of \(A\), the subspace of vectors that are orthogonal to every row of \(A\). When \(m=2\) and \(n=3\), for example, one can understand geometrically why this space must be a \(1\)-dimensional line through the origin in \(\R^3\), along a normal vector to the plane spanned by the rows. The principle is the same in the general case.
We emphasize that the cross product of two vectors is a vector, whereas the dot product of two vectors is a real number.
One can check by elementary computations that these two definitions are equivalent by showing the algebraic version has the following properties:
It is orthogonal to both \(\mathbf a\) and \(\mathbf b\).
Hint
Show this by computing \(\mathbf a \cdot (\mathbf a \times \mathbf b)\) and \(\mathbf b \cdot (\mathbf a \times \mathbf b)\).
\(|\mathbf a\times\mathbf b|^2 = |\mathbf a|^2 |\mathbf b|^2 - (\mathbf a\cdot \mathbf b)^2\)
\(\det [{\bf a, b, a\times b}]>0\)
Hint
Show by expanding the determinant that \[\det [{\bf a, b, a\times b}]=(a_2b_3 - a_3b_2)^2+ (a_3b_1 - a_1b_3)^2+ (a_1b_2 - a_2b_1)^2.\]
Some useful algebraic properties of the cross product are:
The cross-product is not associative; you should try to find examples of vectors \(\mathbf a, \mathbf b, {\bf c}\) such that \[ (\mathbf a\times \mathbf b)\times {\bf c} \ne \mathbf a\times (\mathbf b \times {\bf c}). \]
\(\Leftarrow\) \(\Uparrow\) \(\Rightarrow\)
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