Sometimes we need to index a set of elements of a v ector. In this case, we. W e use the.
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F or example. W e usually giv e matrices upp ercase.
If a real-v alued matrix. W e usually. F or example,. W e can iden tify. This is known as the. Lik ewise,. When we need to explicitly iden tify the elements of. Sometimes we may need to index matrix-v alued expressions that are not just. In this case, w e use subscripts after the expression but do not. In the general case, an array of num b ers arranged on a regular grid with a.
W e identify the element of. One imp ortant op eration on matrices is the. The transp ose of a. W e denote the transp ose of a. V ectors can b e thought of as matrices that contain only one column. Sometimes we. Figure 2. A scalar can b e thought of as a matrix with only a single en try. F rom this, we. W e can add matrices to each other, as long as they hav e the same shap e, just. W e can also add a scalar to a matrix or m ultiply a matrix b y a scalar, just. In the con text of deep learning, we also use some less conv entional notation.
In other words, the v ector. This implicit cop ying of.
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One of the most imp ortant op erations inv olving matrices is multiplication of tw o. W e can write the matrix product just by placing tw o or more matrices. Note that the standard pro duct of tw o matrices is not just a matrix containing. Such an op eration exists and is called the.
W e can think of the matrix pro duct. Matrix pro duct op erations hav e man y useful prop erties that make mathematical. F or example, matrix m ultiplication is. It is also asso ciative:. Matrix multiplication is not comm utativ e the condition. Ho wev er, the dot pro duct b etw een tw o. The transp ose of a matrix pro duct has a simple form:. This enables us to demonstrate equation 2.
Since the fo cus of this textb o ok is not linear algebra, we do not attempt to. W e no w kno w enough linear algebra notation to write down a system of linear. Each element. Each row of. W e can rewrite equation 2. Matrix-v ector pro duct notation provides a more compact represen tation for. An identit y matrix is a matrix that do es not c hange an y v ector when we.
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W e denote the iden tit y matrix that preserves. F ormally , I. The structure of the identit y matrix is simple: all the en tries along the main. W e can now solve equation 2.
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W e discuss. It is not p ossible, how ever, to hav e more than. T o analyze how many solutions the equation has, think of the columns of. In this view, each element of. In general, this kind of operation is called a. F ormally , a. This particular span is known as the. In order for the system. The requirement that the column space of. Otherwise, the. F or example, consider a.
The target. The equation has a solution. Consider a 2. This has the same column space as a 2. In other words, the column space is still just a line and fails to. F ormally , this kind of redundancy is known as. A set of.
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If we add a v ector to a set that is a linear combination of. This means that for the column space of the matrix to encompass all of. Note that the requirement is for a set to hav e exactly. No set of. F or the matrix to hav e an inv erse, w e additionally need to ensure that equa-.
Otherwise there is more than one. T ogether, this means that the matrix m ust b e.