Four fundamental subspaces( for matrix A)   if A is m by n matrix: Column space  C(A) in Rm (列空间在m维实空间中) Null space N(A) in Rn Row space C(A^)(^代表转置)in Rn (all combinations of rows=all columns of A^) Null space of A^ N(A^) in Rm  (left null space of…

Professor: Gilbert Strang Text: Introduction to Linear Algebra http://web.mit.edu/18.06   Lecture 1 contents: n linear equation, n unknowns Row picture & Column picture Matrix form   引入方程组 可表示为AX=b的形式,为: 从几何意义上理解,每个方程表示一条直线,两条直线相交于一点,即为方程组的解.以列的形式可以写…

Linear independence Spanning a space Basis and dimension 以上概念都是针对a bunch of vectors, 不是矩阵里的概念   Suppose A is m by n with m<n, then there are non-zero solutions to AX=0(more unknowns than equations) Reason: There will be free variables Independence: V…

Compute solution of AX=b (X=Xp+Xn) rank r r=m solutions exist r=n solutions unique   example: 若想方程有解,b1,b2,b3需要满足什么条件? 观察矩阵可知,第三行是前两行的和,所以b1+b2=b3 Solvability Condition on b: Ax=b is solvable when b is in C (A) If a combination of Rows of A gives zer…

Computing the nullspace (Ax=0) Pivot variables-free variables Special solutions: rref( A)=R   rank of A=the number of pivots=2 由上述矩阵行变换回代可得方程 我们自行给free variable对应的x赋值 得到一个特解x为 表示的意思是   再赋一组值 表示的意思是 以上特解可以组成通解   Reduce row(echelon form:zeros above and…

Vector spaces and subspaces Column space of A solving Ax=b Null space of A   Vector space requirements v+w and cv are in the space All combs cv+dw are in the space 向量空间对数乘和加法需要封闭 subspace of R^3: Line( L) through zero vector  is a subspace of R^3 Pla…

Section 2.7     PA=LU and Section 3.1   Vector Spaces and Subspaces   Transpose(转置) example: 特殊情况,对称矩阵(symmetric matrices),例如: 思考:R^R(R的转置乘以R)有什么特殊的? 回答:always symmetric why?   Permutation(置换) P=execute row exchanges 之前A=LU是建立在no row exchanges 的基础上的,…

Inverse of AB,A^(A的转置) Product of elimination matrices  A=LU (no row exchanges)   Inverse of AB,A^(A的转置):   Product of elimination matrices  A=LU (no row exchanges) E32E31E21A=U (no row exchanges)    EA=U A=E21`E31`E32`U L表示下三角矩阵,lower triangle D表示对角…

Matrix multiplication(4 ways!) Inverse of A Gauss-Jordan / find inverse of A   Matrix multiplication 1.点积法 2.matrix * column=comb of columns columns of C are comb of cols of A 3.matrix * row = comb of rows rows of C are comb of rows of B 4.matrix * m…

B. Linear Algebra Test   time limit per test 3.0 s memory limit per test 256 MB input standard input output standard output Dr. Wail is preparing for today's test in linear algebra course. The test's subject is Matrices Multiplication. Dr. Wail has n…