Linear algebra problem: given that matrix A is m * n, how to prove that R (AB) = R (BA) = R (a)? Where B is the transpose matrix of matrix A

Linear algebra problem: given that matrix A is m * n, how to prove that R (AB) = R (BA) = R (a)? Where B is the transpose matrix of matrix A

If a is a real matrix
To prove rank (a ^ TA) = rank (a), we only need to verify that a ^ tax = 0 and Ax = 0 have the same solution (note that a ^ tax = 0 = > (AX) ^ tax = 0)

Linear algebra. To prove that B is the inverse of a, must we prove AB = Ba = e, or only prove AB = e

According to the definition of invertible matrix: let a be a matrix of order n, if there is a matrix B of order n such that ab = Ba = e holds, then a is an invertible matrix. Theorem: if a is a matrix of order n and satisfies AB = e, then there must be ba = E. according to the definition of invertible matrix, if AB = Ba = e, then a is an invertible matrix and B is the inverse matrix of A

The sum of the circumference of the two circles is 94.2 cm. It is known that the radius of the big circle is the radius of the small circle, and the ratio of the radius is 4:1. What are the square percentages of the area of the two circles

Circumference: 94.2 times 4 / 5 = 75.36cm
The square of (75.36 divided by 3.14 divided by 2) multiplied by 3.14 = 452.16 square centimeter
Circumference of small circle: 94.2 times 1 / 5 = 18.84cm
The square of (18.84 divided by 3.14 divided by 2) multiplied by 3.14 = 28.26 square centimeter

Take the sides AB and AC of △ ABC as the sides, make square ABDE and square acfg outward respectively, connect eg, try to judge the relationship between △ ABC and △ AEG area, and

(1) The area of △ ABC is equal to that of △ AEG. Reason: if cm ⊥ AB is set at m, G is set at GN ⊥ EA, and EA extension line is set at n, then ∠ AMC = ∠ ang = 90 ° ∫ Quad ABDE and Quad acfg are both square ∫ BAE = ∠ CAG = 90 °, ab = AE, AC = Ag ∫ BAC + ∠ EAG = 180 °∫ EAG + ∠ Gan = 180 °

Bow area formula chord length and height
It is known that the chord length of the bow is 8.8, and the highest position is 1.4. To find the area, write down the formula

Analysis: because 8.8 > 1.4 π, so this bow is a bad arc bow, the answer is as follows: draw a picture, construct a sector, the area of the bow is equal to the sector area minus the triangle area. Known: chord length m, chord height h (that is, the highest position you say) suppose: the radius of the circle where the bow is located is r, then there is (M /...) in the right triangle in the graph

Xiao Ming added one of the numbers one more time when he added the continuous natural number from the beginning. The sum is 149. What is this number?
Be sure to explain the reason

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16=136
149-136=13
The number is 13

Appreciation of the second natural paragraph of from BaiCaoYuan to Sanwei Bookstore

The second paragraph is well-organized. Needless to say, it does not need to say. It summarizes the interesting scenery and things in the herb garden. Here is to highlight the following "Shan Shi."
The first "don't have to say" writes about still life from low to high, and the second "don't have to say" writes about animals from high to low
The combination of these sequences not only describes children's curious eyes, but also makes the scenery conform to people's observation order

Some problems of inverse proportion function in mathematics of grade two in junior high school
1. When the function value of inverse scale function y = (A-3) multiplied by (a square of x-2a-4) is 4, the value of independent variable x is ()
2. If one of the intersections of the functions y = 4x and y = 1 / X is (1 / 2,2), then the coordinate of the other intersection is ()
3. Given y = Y1 + Y2, where Y1 and X are in positive proportion, Y2 and the square of X are in inverse proportion, and when x = 2 and x = 3, the value of Y is equal to 19, find the functional relationship between Y and X

2. From y = 4x and y = 1 / x, we get 4x = 1 / x, 4x & sup2; = 1, x = ± 1 / 2. Another intersection coordinate (- 1 / 2, - 2) is in the third quadrant. 3. Let Y1 = K1X, y2 = K2 / X & sup2; have y = K1X + K2 / X. substituting x = 2, y = 19 and x = 3, y = 19 into: 19 = 2K1 + K2 / 4 (1) 19 = 3k1 + K2 / 9 (2) respectively, we can get K1 = 5, K2 = 36, so y = 5

The bottom of a cuboid is a square with a circumference of 12 decimeters, and its side expansion is just a square. Find the surface area and volume of the cuboid
Remember to calculate the surface area and volume! Two! Answer in one minute,

Cuboid length = width = 12 △ 4 = 3 decimeters
Height = 12 decimeters,
The surface area of the cuboid is 3 × 3 × 2 + 3 × 12 × 4 = 162 square decimeters
Volume = 3 × 3 × 12 = 108 cubic decimeter

Given sin (π 4 − x) = 35, the value of sin2x is ()
A. 1925B. 1625C. 1425D. 725

Method 1: from the known 22 (cosx − SiNx) = 35, the square of both sides is 12 (1 − sin2x) = 925, and then get sin2x = 725; method 2: let π 4 − x = α, then sin α = 35, so sin2x = sin (π 2 − 2 α) = Cos2 α = 1 − 2sin2 α = 725